Mustapha Mokhtar-Kharroubi
email: mmokhtar@univ-fcomte.fr

Compactness properties of perturbed sub-stochastic semigroups on L 1 ( ): A preliminary version

We deal with positive c 0 -semigroups fU (t); t > 0g of contractions in L 1 ( ; A; ) with generator T where ( ; A; ) is an abstract measure space and provide a systematic approach of compactness properties of perturbed semigroups e t("T V ") ; t > 0 (or their generators) induced by singular and bounded below potentials V : ( ; ) ! R. The results are precised further for metric measure spaces ( ; d; ). This new theory relies on several ingredients: new a priori estimates peculiar to L 1 -spaces, local weak compactness assumptions on unperturbed operators, "Dunford-Pettis" arguments and the assumption that the sublevel sets M := fx; V (x) M g are "thin at in…nity with respect to fU (t); t > 0g". We show also how spectral gaps occur when the sublevel sets are not "thin at in…nity". This formalism suits c 0 -semigroups with integral kernels, for instance most sub-markovian semigroups arising in the theory of Markov processes in general state spaces, and combines intimately the kernel of fU (t); t > 0g and the sublevel sets M . Inde…nite potentials are also dealt with. We illustrate the relevance of some aspects of this theory by giving new compactness and spectral results on convolution semigroups, magnetic Schrödinger semigroups, weighted Laplacians (in particular the Poincaré inequality for probability measures e (x) dx on R N ) and Witten Laplacians on 1-forms.

Introduction

We deal with perturbed semigroups in L 1 ( ) spaces generated by

T V 1
where T is the generator of a substochastic semigroup and V is a singular inde…nite potential; the meaning of "T V " will be explained below. The object of this paper is to give new functional analytic tools and results on spectral theory (full discreteness or spectral gaps) of perturbed semigroups or perturbed generators. In particular, we are concerned with resolvent compactness of T V and also with existence of spectral gaps for perturbed generators, i.e.

s ess (T V ) < s(T V )

where s(T V ) := sup fRe ; 2 (T V )g

is the spectral bound of T V and s ess (T V ) := sup fRe ; 2 ess (T V )g

is the essential spectral bound of T V ; we note that s(T V ) 2 (T V ) by standard theory of positive semigroups. We study also weak spectral gaps of perturbed generators, i.e. when the peripheral spectrum of T V;

(T V ) \ fs(T V ) + iRg ; consists of isolated eigenvalues with …nite algebraic multiplicities; this spectral picture does not prevent a priori the existence of sequences of eigenvalues of T V with real parts going to s(T V ) and imaginary parts going to in…nity; note that the spectrum (T V ) need not be real.

Similarly, we study the compactness of perturbed semigroups e t(T V ) ; t > 0 and also the existence of spectral gaps, i.e. r ess (e t(T V ) ) < r (e t(T V ) )

where r (e t(T V ) ) is the spectral radius of e t(T V ) and r ess (e t(T V ) ) := sup n j j ; 2 ess (e t(T V ) ) o is the essential spectral radius of e t(T V ) ; ( ess refers to essential spectrum).

We are mainly interested in the situation where a priori T is not resolvent compact and has no spectral gap, i.e. full discreteness or spectral gaps are induced by the presence of a potential V . A new and general theory dealing with many aspects is provided. While most of the known literature on full discretenes or spectral gaps is concerned with hilbertian results and (quite often) by self-adjoint semigroups, we provide here a new point of view relying on a new circle of ideas peculiar to L 1 -spaces and without any connection with self-adjointess. A systematic approach of the underlying compactness background is given. We show also how this general formalism …ts with substochastic semigroups arising in the theory of Markov processes in metric spaces; in particular, various examples from Stastistical Mechanics are dealt with. In the case where e tT ; t > 0 operates on all L p ( ; ) spaces (e.g. for sub-Markov semigroups) then so does e t(T V ) ; t > 0 ; we show then how the L 1 spectral structure of e t(T V ) ; t > 0 determines its L p spectral structure.

The particular role of positive operators and L 1 -spaces in linear perturbation theory appeared a long time ago in the classical Kato's paper [START_REF] Kato | On the semi-groups generated by Kolmogoro¤'s di¤erential equations[END_REF] on well-posedness of Kolmogorov's di¤erential equations and also later in the important analytical and probabilistic role played by the so-called Kato class potentials used currently in the theory of Markov processes, see e.g. [START_REF] Aizenman | Brownian motion and Harnack's inequality for Schrödinger operators[END_REF][77] [START_REF] Voigt | Absorption semigroups, their generators and Schrödinger semigroups[END_REF] [START_REF] Demuth | Stochastic Spectral Theory for Self-Adjoint Feller Operators-A Functional Integration Approach[END_REF]. Recently, essentially from the beginning of the 2000's, there has been a renewal of interest in perturbation theory of substochastic semigroups in L 1 -spaces where the generator of a substochastic semigroup e tT ; t > 0 is perturbed additively by a positive T -bounded operator V (see [START_REF] Banasiak | Perturbations of Positive Semigroups with Applications[END_REF] and references therein); this so-called "honesty theory"is motivated by various problems from fragmentation theory or kinetic theory and has a probabilistic counterpart in the concept of non-explosive processes [START_REF] Tyran-Kaminska | Substochastic semigroups and densities of piecewise deterministic Markov processes[END_REF]. More recent developments on "honesty theory" are given in [START_REF] Mokhtar-Kharroubi | On honesty of perturbed substochastic C 0 -semigroups in L 1 -spaces[END_REF] while a non-commutative version of [START_REF] Mokhtar-Kharroubi | On honesty of perturbed substochastic C 0 -semigroups in L 1 -spaces[END_REF] (in the Banach space of trace class operators in a Hilbert space), of interest for quantum dynamical semigroups (see e.g. [START_REF] Davies | Quantum dynamical semigroups and the neutron di¤usion equation[END_REF] and [START_REF] Fagnola | Quantum Markov semigroups and quantum ‡ows[END_REF]), is given in [START_REF] Mokhtar-Kharroubi | On perturbed positive semigroups on the Banach space of trace class operators[END_REF]; an ultimate extension (with new developments) to general ordered Banach spaces with additive norm on the positive cone is given in [START_REF] Arlotti | On perturbed substochastic semigroups in abstract state spaces[END_REF]. In another direction, it was realized in [START_REF] Mokhtar-Kharroubi | On Schrödinger semigroups and related topics[END_REF] [START_REF] Mokhtar-Kharroubi | Perturbation theory for convolution semigroups[END_REF] [START_REF] Mokhtar-Kharroubi | New form-bound estimates for many-particle Schrödinger-type Hamiltonians[END_REF] that the use of weak compactness arguments in L 1 allows a signi…cant generalisation of the (extended) Kato class potentials for Schrödinger-type operators and provide also new (hilbertian) form-bound estimates; these ideas have also useful applications to kinetic theory [START_REF] Mokhtar-Kharroubi | New generation theorems in transport theory[END_REF]. The main goal of the present paper is to show how L 1 weak compactness arguments, combined to new L 1 estimates, allow a systematic spectral analysis of perturbed substochastic semigroups in the case of negative unbounded multiplication operators V ; we show also how to combine those ideas to another ones (inspired by transport theory [START_REF] Mokhtar-Kharroubi | Mathematical Topics in Neutron Transport Theory[END_REF]) to cover also inde…nite potentials V = V + V ; in both situations, local weak compactness tools play a key role. Thus, following the spirit of [START_REF] Mokhtar-Kharroubi | On Schrödinger semigroups and related topics[END_REF][60] [START_REF] Mokhtar-Kharroubi | New form-bound estimates for many-particle Schrödinger-type Hamiltonians[END_REF], this paper continues the exploration of the role of the space L 1 and its weak topology in well-posedness of perturbed evolu-tion equations, their spectral analysis and also their relevant applications to various sub-markovian equations arising e.g. in classical probability theory.

Let ( ; A; ) be a measure space and let fU (t); t > 0g be a positive c 0semigroup of contractions on L 1 ( ; A; ) with generator T: We denote by V : ( ; ) ! [0; +1] (or more generally bounded from below) a measurable potential and denote by fU V (t); t > 0g the (appropriately de…ned) perturbed semigroup generated by T V :="T V ". The main object of this paper is to give a general and systematic theory of full spectral discreteness or spectral gaps for perturbed generators T V :="T V " or perturbed semigroups fU V (t); t > 0g and to illustrate this theory by signi…cant examples of applied interest. More precisely, we focus on the underlying compactness background. The interplay between the singular potential and the unperturbed semigroup which is in the heart of such compactness or spectral gaps results is …nely analyzed in this paper. We give here a point of view on the subject relying on new tools peculiar to L 1 spaces; a completely new formalism is provided. In our general context, the relevant technical tools we need will be di¤erent depending on whether we deal with T V or fU V (t); t > 0g : Thus, in our study of perturbed generators T V :="T V ", we take advantage of the quite unsuspected fact, in comparison to L 2 -space setting, that V is always T V -bounded in L 1 spaces [START_REF] Oinarov | On the separability of the Schrödinger operator in the space of summability functions[END_REF] [START_REF] Voigt | Absorption semigroups, their generators and Schrödinger semigroups[END_REF]. On the other hand, to study perturbed semigroups fU V (t); t > 0g, we provide two di¤erent strategies: the …rst approach consists in assuming that fU (t); t > 0g is norm continuous, in showing the norm continuity of the perturbed semigroup fU V (t); t > 0g and in taking advantage of the properties of perturbed generators and "spectral mapping tools". In the second strategy, we show a "weak type" estimate for almost all t > 0 Z

fV >M g

(U V (t)f ) (dx) c t kf k M ; 8 f 2 L 1 + ( ; ); 8 M > 0
under the assumption (on the perturbed semigroup) that 8t > 0; sup

"2]0;1] U V (t + ")1 U V (t)1 " L 1 ( ; ) < +1
where U V (t) is the dual operator of U V (t). (This last assumption is much weaker than a di¤erentiability condition on fU V (t); t > 0g and is satis…ed e.g. if

8f 2 L 1 ( ; ); ]0; +1[ 3 t ! Z U V (t)f is di¤erentiable or if ]0; +1[ 3 t ! U V (t) 2 L(L 1 ( ; )) is locally lipschitz;
in particular it is satis…ed if fU (t); t > 0g is holomorphic because fU V (t); t > 0g is then holomorphic too [START_REF] Arendt | Absorption semigroups and Dirichlet boundary conditions[END_REF] [START_REF] Ishikawa | Analyticity of absorption semigroups[END_REF].) These L 1 -estimates combined to local weak compactness assumptions on unperturbed operators, to properties of sublevel sets M := fy; V (y) M g ; more precisely their "size at in…nity with respect to unperturbed operators", and to "Dunford-Pettis"arguments, play an important part in our formalism and provide us with new relevant tools in spectral theory of perturbed substochastic semigroups and their generators. Our local L 1 weak compactness assumptions on unperturbed operators are very weak ones and are trivially satis…ed by most examples occuring in the literature.

We also deal with inde…nite potentials V = V + V (with nonnegative V ), i.e. with operators T (V + V ) considered as perturbed operators

T V + + V :
This second perturbation theory combines the previous one and di¤erent ideas inspired by transport theory [START_REF] Mokhtar-Kharroubi | Mathematical Topics in Neutron Transport Theory[END_REF].

Before explaining more precisely the content of this work, some related information in Hilbert space setting is worth mentioning. According to a classical result going back at least to K. Friedrichs [START_REF] Friedrichs | Spektraltheorie halbbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Di¤erential operatoren[END_REF], Schrödinger operators + V in L 2 (R N ) (de…ned by means of quadratic forms) have fully discrete spectra, or equivalently + V has a compact resolvent, for nonnegative potentials V 2 L 1 loc (R N ) such that lim jxj!1 V (x) = +1: Of course, it is also known since a long time that this condition is not necessary since F. Rellich [START_REF] Rellich | Das Eigenwertproblem von 4u + u = 0 in Halbröhren, in: Studies and Essays Presented to R[END_REF] already observed for example that for the potential V (x 1 ; x 2 ) = x 2 1 x 2 2 ;

(1)

+ V is still resolvent compact in L 2 (R 2 ) even if V (x 1 ;
x 2 ) fails to go to +1 at in…nity near the axes. Besides K. Friedrichs [START_REF] Friedrichs | Spektraltheorie halbbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Di¤erential operatoren[END_REF], the literature on discreteness of the spectrum of Schrödinger operators goes back to A.M. Molchanov [START_REF] Molchanov | The conditions for the discreteness of the spectrum of self-adjoint second order di¤erential equations[END_REF] and is now considerable; we refer to the survey [START_REF] Shubin | Spectral theory of the Schrödinger operators on noncompact manifolds: qualitative results[END_REF] and also to the more recent paper [START_REF] Maz | Discreteness of spectrum and positivity criteria for Schrödinger operators[END_REF] for more developments. This literature deals with Schrödinger operators on more general non-compact Riemannian manifolds and provides optimal (i.e. necessary and su¢ cient) conditions of discreteness in terms of Wiener capacity of suitable sets. Such sharp results are not always of simple practical use but su¢ cient or necessary conditions in terms of measures are also available; among the various statements we note A.M. Molchanov's necessary condition of discreteness Z B(x;r)

V (y)dy ! +1 as x ! 1 (2)
(which is also su¢ cient in one dimension) and also the su¢ cient criterion:

Theorem 1 ([74] Corollary 10.2, p. 268). We assume that for any M > 0 the sublevel set M := fy; V (y) M g is "thin at in…nity" in the sense that for some r > 0 jB(x; r) \ M j ! 0 as x ! 1

where B(x; r) is the ball centered at x with radius r (and jj refers to Lebesgue measure). Then

+ V in L 2 (R N ) has a discrete spectrum.
In ( [START_REF] Georgescu | Hamiltonian with purely discrete spectrum[END_REF] Lemma 5 and Remark 2) it is observed that the sublevel sets of a nonnegative function V are "thin at in…nity" if and only if Z B(x;r)

1 1 + V (y) dy ! 0 as x ! 1; (4)
the argument relies on the simple double inequality (for arbitrary M > 0)

1 1 + M jB(x; r) \ M j Z B(x;r) 1 1 + V (y) dy Z B(x;r) 1 1 + V (y) dy jB(x; r) \ M j + 1 1 + M jB(0; r)j :
One realizes then that Theorem 1 was already known in 1978 under Assumption (4) [START_REF] Benci | Discreteness Conditions of the Spectrum of Schrödinger Operators[END_REF]; it seems that this has not been noticed in the literature on the subject. One sees also how the necessary condition (2) follows from "thinness at in…nity" of sublevel sets M since

jB(0; r)j = jB(x; r)j = Z B(x;r) p 1 + V (y) p 1 + V (y) dy ( Z B(x;r) 1 1 + V (y) dy) 1 2 ( Z B(x;r) (1 + V (y))dy) 1 2
and then Z

B(x;r) V (y)dy > jB(0; r)j + jB(0; r)j 2 R B(x;r) 1 1+V (y) dy : More recently, it was shown in [48] that T V is resolvent compact in L 2 (R N ) when T is the relativistic -stable operator T = ( + m 2 ) 2 + m
provided that lim jxj!1 V (x) = +1. This result was extended in [START_REF] Wang | Compactness of Schrödinger semigroups with unbounded below potentials[END_REF] (for sublevels sets M having …nite measure only) to much more general symmetric Markov generators in L 2 ( ; ) satisfying the so-called intrinsic super Poincaré inequality and such that the Markov semigroup has a density with respect to . The proof given by the authors is however quite involved and combines various technical arguments; shortly after, a simpler proof was given in [START_REF] Simon | Schrödinger operators with purely discrete spectrum[END_REF] and other developments, still for self-adjoint operators in Hilbert spaces, were also given in [28][51]. Even the …niteness assumption on the measure of the sublevels sets M has been dropped. For instance, we …nd in [START_REF] Simon | Schrödinger operators with purely discrete spectrum[END_REF] that if T is a self-adjoint operator in L 2 ( ; ) such that e tT ; t > 0 is an ultracontractive semigroup in the sense that (for t > 0)

e tT maps L 2 ( ; ) into L 1 ( ; ) then T V is resolvent compact in L 2 ( ; ) provided that V 2 L 1 loc (R N
) and V > 0 is such that its sublevels sets are r-polynomially thin (for some r > 0), i.e. for any R > 0 Z M j M \ B(x; R)j r (dx) < +1:

We note that in R N , r-polynomially thin set is necessarily thin at in…nity in the sense (3) (see [START_REF] Georgescu | Hamiltonian with purely discrete spectrum[END_REF] Lemma 7). On the other hand, it is known (see e.g. [START_REF] Avron | Schrödinger operators with magnetic …elds. I General interactions[END_REF]) that the discreteness of the spectrum of the magnetic Schrödinger operator in L 2 (R N ) is strongly connected to that of the Schrödinger operator via the diamagnetic inequality. Finally, there exists also an important literature on Poincaré (or spectral gap) inequalities for Markov semigroups arising in Probability and Stastistical Mechanics

var (f ) := Z f 2 d ( Z f d ) 2 c(A 1 2 f; A 1 2 f ); f 2 D(A 1
2 );

of interest e.g. for exponential trend to equilibrium, where ( ; ) is a probability space, A is a nonnegative self-adjoint operator in L 2 ( ; ), 1 2 D(A) and A1 = 0; (such inequalities are sometimes derived from Log Sobolev (or Gross) inequalities; see [START_REF] Gross | Logarithmic Sobolev Inequalities and contractivity properties of semi-groups[END_REF][72] [3][37][85]). Note that a spectral gap expresses simply that 0; the bottom of (A), is an isolated eigenvalue with …nite algebraic multiplicity; as such, the notion of a spectral gap is meaningful in much more general situations but, of course, cannot be formulated in terms of variance inequality. Actually, this notion amounts to strict positivity of the bottom of the essential spectrum ess (A); we refer to [68][55] for the location of essential spectra of Schrödinger operators +V in L 2 (R N ) when the sublevel sets of V are not "thin at in…nity". We point out that all the results above are hilbertian; in particular no L 1 compactness result nor spectral gap result in L 1 space can a priori be derived from the literature above. We mention also the paper [START_REF] Gong | Spectral gap of positive operators and applications[END_REF] on spectral gaps for bounded positive operators in L p -spaces with p > 1 and various applications. This very brief overview shows various contexts where full discreteness or spectral gaps are worth studying.

As far as potentials bounded from below are concerned, we provide here a new point of view on discretenes and on spectral gaps relying on a di¤erent circle of ideas. Neither selfadjointness nor L 2 spaces play a role in our approach. By contrast, the functional space L 1 ( ; ); the positivity of the unperturbed semigroup under consideration (this could be relaxed by relying on its modulus [START_REF] Kipnis | Majoration des semigroupes de contraction de L 1 et applications[END_REF], see Section 6), the fact that V is always T V -bounded in L 1 spaces and the "weak type" estimate

Z fV >M g (U V (t)f ) (dx) c t kf k M ; 8 f 2 L 1 + ( ; ); 8 M > 0;
valid under suitable assumptions (or the norm continuity of fU V (t); t > 0g valid under other suitable assumptions) provide us with the starting point of a completely new formalism. By complementing these L 1 -estimates by local weak compactness assumptions on ( T ) 1 or fU (t); t > 0g and taking advantage of "Dunford-Pettis" arguments, we can build a general theory where various related functional analytic results are given; the stability of essential spectra by weakly compact perturbations (see e.g. [START_REF] Latrach | Essential Spectra on Spaces with the Dunford-Pettis Property[END_REF]), combined to the above ingredients, turns out to be the right tool to deal with spectral gaps. We provide thus a pure L 1 theory on full discretenes or spectral gaps where most of the results are new. Moreover, our construction has the advantage of being conceptually simple and self-contained. We note that in the special case where the semigroup operates in all L p spaces (1 p < +1), e.g. for sub-Markov semigroups, our L 1 compactness results imply, by interpolation, compactness results in L p for p > 1; providing us e.g. with hilbertian results, while converse statements are not true a priori, see e.g. Markov semigroups generated by weighted Laplacians which are compact in L p for p > 1 but fail to be so in L 1 [18] Section 4.3. We point out that our primary goal here is not a priori to obtain hilbertian results by means of L 1 techniques; it is rather to build and explore an L 1 theory for its own sake and this program is undertaken here for the …rst time. We note also that our L p results (p > 1) are new since they are deduced from an L 1 theory; in a sense, L p spectral theory of perturbed submarkovian semigroups becomes a sub-product of the L 1 theory which acquires thus a special status. Our results are given for general measure spaces ( ; A; ) and precised further for metric measure spaces ( ; d; ). In addition, a special section is also devoted to more speci…c results on convolution semigroups on euclidean spaces (e.g. on subordinate Brownian semigroups) because of their importance in applications. Finally, various related examples from Statistical Mechanics are revisited: thus, Markov semigroups steming from Dirichlet forms in weighted L 2 spaces turn out to be unitarily equivalent to usual Schrödinger semigroups (with potentials) that can be dealt with by our L 1 formalism while various spectral results on Witten Laplacians on 1-forms (of interest for Hel¤er-Sjöstrand's covariance formula) are also given. Our approach of the subject suits semigroups exhibiting integral kernels; this happens under our local weak-compactness assumptions (see Remark 15 (ii)); e.g. ultracontractive symmetric Markov semigroups for separable measure spaces meet our conditions. Thus, for L 1 spaces over metric measure spaces ( ; d; ), our compactness or spectral gap results combine intimately integral kernels of unperturbed semigroups and sublevel sets M := fy; V (y) M g of the singular potential. For instance, this provides us with su¢ cient conditions in terms of heat kernel and sublevel sets of 1 4 jr j 2 1 2 4 ) for a probability measure e (x) dx on R N to satisfy the Poincaré inequality. A last section is devoted to inde…nite potentials

V = V + V :
Actually, all the paper shows how fruitful is the L 1 treatment of perturbed sub-Markov semigroups provided some reasonable upper estimate on their integral kernels is available.

Of course, various kinds of upper estimates of transition kernels appear in the literature on Markov processes in metric spaces. For instance, the Heat kernel associated to the Laplace Beltrami operator on non-compact complete Riemannian manifolds ( ; d; ) of dimension n (d is the geodesic distance and is the Riemannian volume) with Ricci curvature bounded below and having the so-called "bounded geometry"(see [START_REF] Davies | Heat Kernels and Spectral Theory[END_REF] p. 172) satis…es a Gaussian estimate for each t > 0

p t (x; y) C 1 t exp( d(x; y) 2 C 2 t ); (5)
see e.g. [START_REF] Davies | Heat Kernels and Spectral Theory[END_REF][30] [START_REF] Grigor'yan | Heat kernels on weighted manifolds and applications[END_REF]. However, Brownian motions on some fractal spaces lead to transition kernels with sub-Gaussian estimates

p t (x; y) C t exp( ( d (x; y) Ct ) 1 1 ) (6)
where > 0 is the Hausdor¤ dimension and > 2 is "a walk dimension", see e.g. [START_REF] Barlow | Brownian motion and harmonic analysis on Sierpinski carpets[END_REF]. On the other hand, the study of kernel estimates for non local Dirichlet forms, in connection with Markov processes with jumps, developped also in the last decades and typical kernel estimates of jump Markov semigroups are polynomial

p t (x; y) C t (1 + d(x; y) t 1 ) ( + ) ; (7)
see e.g. [START_REF] Hu | Nash type inequalities and Heat kernels for non local Dirichlets forms[END_REF]. (We refer to [7][32] for much more information on the very rich subject of "Heat kernels".) This "ubiquity"of integral kernels suggests that there is a room for a general theory of compactness (and spectral) properties for a large class of perturbed c 0 -semigroups which is the object of this work.

We outline now some of our main results:

In Section 2, we consider a measure space ( ; A; ) and a positive c 0semigroup of contractions fU (t); t > 0g on L 1 ( ; A; ) with generator T: Let V : ( ; ) ! [0; +1] be measurable and let fU V (t); t > 0g be the contraction semigroup de…ned by U V (t)f := lim n!+1 e t(T Vn) f where V n := V ^n: We note that this semigroup need not a priori be strongly continuous at the origin but we restrict ourselves to the case where it is so and denote by T V its generator. We show that T V has a compact resolvent if and only if for all M > 0 the operator

( T V ) 1 : L 1 ( ; ) ! L 1 ( M ;
) is weakly compact (we mean the operator:

f 2 L 1 ( ; ) ! ( T V ) 1 f j M 2 L 1 ( M ;
)) where M := fy; V (y) M g are the sublevel sets of V . It follows from the domination ( T V ) 1 ( T ) 1 that a su¢ cient condition for T V to be resolvent compact it that

( T ) 1 : L 1 ( ; ) ! L 1 ( M ; ) is weakly compact; ( 8
)
under a technical additional assumption (which is satis…ed e.g. in denumerable state spaces), we show that (8) is also necessary.

If

fU V (t); t > 0g is norm continuous i.e. ]0; +1[ 3 t ! U V (t) 2 L(L 1 ( ; ))
is continuous in operator norm, then (8) implies the stronger result that the perturbed semigroup fU V (t); t > 0g is compact on L 1 ( ; ). The question about when this norm continuity assumption is satis…ed is also dealt with: we show …rst the stability estimate

sup t C e t(T Vn) f U V (t)f e C [V V n ] (1 T V ) 1 f ; 8f 2 L 1 + ( ; )
for arbitrary C > 0 (where

[V V n ] (1 T V ) 1
n is a sequence of bounded operators going strongly to zero as n ! +1) which has its own interest and which implies that fU V (t); t > 0g is norm continuous provided that fU (t); t > 0g is norm continuous and

[V V n ] (1 T V ) 1 L(L 1 ( ; )) ! 0 as n ! +1;
in particular, if (1 T V ) 1 is an integral operator with kernel G V (x; y) then fU V (t); t > 0g is norm continuous provided that fU (t); t > 0g is norm continuous and

sup y2 Z fV >ng G V (x; y)V (x) (dx) ! 0 as n ! +1: (9)
On the other hand, if for all M > 0

U (t) : L 1 ( ; ) ! L 1 ( M ; ) is weakly compact (t > 0) (10)
and if the dual operator U V (t) satis…es 8t > 0; sup

"2]0;1] U V (t + ")1 U V (t)1 " L 1 ( ; ) < +1 (11)
(the latter holds e.g. if

8f 2 L 1 ( ; ); ]0; +1[ 3 t ! Z U V (t)f is di¤erentiable or if ]0; +1[ 3 t ! U V (t) 2 L(L
(U V (t)f ) (dx) c t kf k M ; 8 f 2 L 1 + ( ; ); 8 M > 0
which turns out to be a key tool in the study of spectral gaps for fU V (t); t > 0g.

We note that [START_REF] Benci | Discreteness Conditions of the Spectrum of Schrödinger Operators[END_REF] is much weaker than a di¤erentiability condition on the perturbed semigroup fU V (t); t > 0g and does not even imply its norm continuity, see Remark 4(ii). We show also that T V is resolvent compact if (10) is satis…ed. Moreover, Assumption (10) is shown to be stable by subordination; the proof of this relies on the fact that a strong integral (not necessarily a Bochner integral) on a …nite measure space of a strongly measurable bounded operator-valued mapping with values in W (E; F ) (the space of weakly compact operators between Banach spaces E and F ) belongs to W (E; F ) [START_REF] Schuchtermann | On weakly compact operators[END_REF]; see also [START_REF] Mokhtar-Kharroubi | On the strong convex compactness property for the strong operator topology and related topics[END_REF] when F is an L 1 ( )-space. This result has signi…cant applications e.g. to subordinate Brownian semigroups on euclidean spaces (see below).

In the special case where fU (t); t > 0g operates on all L p ( ; ) spaces then so does fU V (t); t > 0g (we note them respectively fU p (t); t > 0g and fU pV (t); t > 0g when acting in L p ( ; )) and then various compactness results in L p ( ; ) are also obtained by interpolation; in particular, under Assumption (8) only, if fU 2 (t); t > 0g is symmetric in L 2 ( ; ) then fU pV (t); t > 0g is a compact semigroup in L p ( ; ) for p > 1 (but is not a priori so in L 1 ( ; )).

Because of their applied interest, we devote Section 3 to speci…c results on convolution semigroups (related to Lévy processes) on euclidean spaces. We show …rst that if h 2 L 1 (R N ) and if

H : ' 2 L 1 (R N ) ! Z R N h(x y)'(y)dy then, for a Borel set R N , H : L 1 (R N ) ! L 1 ( ) is compact if and only if sup y2R N Z \fjxj>cg h(x y)dx ! 0 as c ! 1
and the latter condition is satis…ed if is "thin at in…nity"in the sense (3). This allows us to deal with convolution semigroups

U (t) : f 2 L 1 (R N ) ! Z f (x y)m t (dy) 2 L 1 (R N )
where fm t g t 0 are Borel sub-probability measures on R N such that m 0 = 0 (the Dirac measure at zero), m t m s = m t+s and m t ! m 0 vaguely as t ! 0 + . The sub-probability measures fm t g t>0 are characterized by

c m t ( ) := (2 ) N 2 Z e i :x m t (dx) = (2 ) N 2 e tF ( ) ; 2 R N
where F ( ) is the so-called characteristic exponent; (see e.g. [START_REF] Jacob | Pseudo-Di¤ erential Operators & Markov Processes[END_REF] Chapter 3). The resolvent of the generator T is also a convolution with a measure

m ( T ) 1 f = Z f (x y)m (dy) where m = R +1 0 e t m t dt ( > 0) is a vaguely convergent integral such that c m ( ) = Z +1 0 e t c m t ( )dt = 1 + F ( ) :
Thus, if m is a function (i.e. is absolutely continuous with respect to Lebesgue measure), in particular if e tF ( ) 2 L 1 (R N ) for t > 0, then T V has a compact resolvent provided that the sublevel sets M are "thin at in…nity" in the sense (3): We show also the compactness of perturbed semigroups in L p (R N ) (p > 1) for all subordinate Brownian semigroups; this covers for example the relativistic -stable semigroup generated by

T = ( + m 2 ) 2 + m (0 < < 2; m > 0):
In Section 4, we deal with sub-stochastic semigroups in L 1 spaces over metric measure spaces, i.e. metric spaces ( ; d) endowed with a Borel measure which is …nite on bounded Borel subsets of . This framework is motivated by Markov processes in metric spaces, (see e.g. [START_REF] Grigor'yan | Heat kernels on metric measure spaces with regular volume growth[END_REF] and references therein). The existence of a metric d allows to precise further some of the results in Section 2. We show that if [START_REF] Benci | Discreteness Conditions of the Spectrum of Schrödinger Operators[END_REF] is satis…ed (e.g. if fU (t); t > 0g is holomorphic) and if U (t) is such that U (t) : L 1 ( ; ) ! L 1 ( ; ) is weakly compact for any bounded Borel set then fU V (t); t > 0g is a compact semigroup in L 1 ( ; ) provided that for some x 0 2 lim

C!+1 sup y2 Z fx2 M ; d(x;x 0 )>Cg p t (x; y) (dx) = 0
where p t (x; y) is the kernel of U (t) (the existence of this kernel is a consequence of the weak compactness assumption, see [START_REF] Dunford | Linear Operators, Part 1: General Theory[END_REF] p. 508); we express this by saying that the sublevel sets M are "thin at in…nity with respect to fU (t); t > 0g". In particular, if v(r) := sup x2 (B(x; r)) < 1 8r > 0 and if p t (:; :) satis…es an estimate of the form

p t (x; y) f t (d(x; y))
where f t : R + ! R + is nonincreasing and such that (for large r) the function r ! f t (r)v(r+1) is nonincreasing and integrable at in…nity then the sublevel sets M are "thin at in…nity with respect to fU (t); t > 0g"if they are "thin at in…nity" in the sense there exists a point y 2 such that for any R > 0 f M \ B(y; R)g ! 0 as d(y; y) ! +1; thus if we consider e.g. the typical examples of kernel estimates (5); ( 6) and [START_REF] Auscher | Heat kernels and Analysis on Manifolds, Graphs and metric Spaces[END_REF] occuring in the study Markov processes in metric spaces, one easily sees which volume growth r ! v(r) (for large r) is compatible with the above assumption on r ! f t (r)v(r + 1): Other statements in terms of the kernel of (1 T ) 1 are also given when fU V (t); t > 0g is norm continuous.

In Section 5, we relax the assumption that the sublevel sets M are "thin at in…nity with respect to fU (t); t > 0g". We show, under suitable kernel estimates involving the sublevel sets M that the essential spectral radius r ess (U V (t)) of the semigroup U V (t)) is less than its spectral radius, i.e. U V (t) exhibits a spectral gap. To this end, we take full advantage of the stability of essential spectra by weakly compact perturbations in L 1 spaces (see e.g. [START_REF] Latrach | Essential Spectra on Spaces with the Dunford-Pettis Property[END_REF]). More precisely, if [START_REF] Benci | Discreteness Conditions of the Spectrum of Schrödinger Operators[END_REF] is satis…ed (e.g. if fU (t); t > 0g is holomorphic) and if for some t > 0, U (t) : L 1 ( ) ! L 1 ( ) is weakly compact for any bounded Borel set and the kernel p t (x; y) of U (t) satis…es the estimate

sup M >0 lim C!+1 sup y2 Z fx2 M ; d(x;x 0 )>Cg p t (x; y) (dx) < e V t
(for some x 0 2 ) where V := s(T V ) is the spectral bound of T V then ! ess < V where ! ess is the essential type of fU V (t); t > 0g; note that e !esst = r ess (U V (t)) 8t > 0; see e.g. [START_REF] Nagel | One-Parameter Semigroups of Positive Operators[END_REF] p. 74. We observe that V is also the type of fU V (t); t > 0g since the latter is a positive semigroup in L 1 space ([65] Theorem 1.1 p. 334) and then

e V t = r (U V (t)) kU V (t)k L(L 1 ( )) kU (t)k L(L 1 ( )) = sup y2 Z p t (x; y) (dx):
One can avoid the use of the (a priori unknown) parameter V and obtain a slightly di¤erent result formulated as an alternative: Indeed, using 1 (the spectral bound of T ) instead of V we show that if

sup M >0 lim C!+1 sup y2 Z fx2 M ; d(x;x 0 )>Cg p t (x; y) (dx) < e 1 t
then either V < 1 or V = 1 and ! ess < V where ! ess is the essential type of fU V (t); t > 0g. In particular, if fU (t); t > 0g is a stochastic semigroup i.e. is mass preserving on the positive cone and if

sup M >0 lim C!+1 sup y2 Z fx2 M ; d(x;x 0 )>Cg p t (x; y) (dx) < 1
then either V < 0 or V = 0 and ! ess < 0: In the case where fU (t); t > 0g operates on all L p ( ) (p > 1) and if we denote by p (resp. pV ) the spectral bound of the generator of fU p (t); t > 0g (resp. of fU pV (t); t > 0g) then, under the estimate

sup M >0 lim C!+1 sup y2 Z fx2 M ; d(x;x 0 )>Cg p t (x; y) (dx) < e p pt ;
we have either pV < p or pV = p and ! pess < pV where ! pess is the essential type of fU pV (t); t > 0g :

We can also avoid such alternatives for symmetric semigroups. We show then that fU V (t); t > 0g has a spectral gap (i.e.

! ess < V ) provided that sup M >0 lim C!+1 sup y2 Z fx2 M ; d(x;x 0 )>Cg p t (x; y) (dx) < e b t where b := lim r!1 b r (increasing limit) b r := inf '2Dr; k'k L 2 ( ; ) =1 p T 2 ' 2 L 2 ( ; ) + Z B(x 0 ;r) V (x) j'(x)j 2 (dx); D := ' 2 D( p T 2 ); Z V (x) j'(x)j 2 (dx) < +1
and D r is the subspace of D consisting of those elements with supports included in B(x 0 ; r). Note that b r b and that the left hand side of the above strict inequality depends on the values of the potential "at in…nity" while the right hand side depends on its values at …nite distance only. We point out that there exists an important literature on spectral gaps in L 2 setting in terms of Poincaré inequalities, (such inequalities are also related to Log Sobolev (or Gross) inequalities, see e.g. [START_REF] Gross | Logarithmic Sobolev Inequalities and contractivity properties of semi-groups[END_REF]

[72][3][37][85][14][29]
). We provide here a new point of view in L 1 spaces in terms of kernels estimates of unperturbed semigroups and sublevel sets of the potential. We also study weak spectral gaps for generators T V . Indeed, we show that if the kernel G 1 (x; y) of (1 T ) 1 satis…es the estimate

sup M >0 lim C!+1 sup y2 Z fx2 M ; d(x;x 0 )>Cg G 1 (x; y) (dx) < 1 1 V (for some x 0 2 ) then the peripheral spectrum of T V , i.e. (T V ) \ f V + iRg ;
consists of isolated eigenvalues with …nite algebraic multiplicities (see Theorem 33 for a more precise statement). This spectral picture does not prevent a priori the existence of sequences of eigenvalues of T V with real parts going to s(T V ) and imaginary parts going to in…nity. However, if fU V (t); t > 0g is norm continuous then we show that above estimate implies the much stronger conclusion that this semigroup has a spectral gap.

Sections 6,7 and 8 illustrate the practical usefulness of some of previous functional analytic results by providing new results in three directions of applied interest: Magnetic Schrödinger operators (Section 6), weighted Laplacians (Section 7) and Witten Laplacians on 1-forms (Section 8); we revisit some important examples of the subject by exploiting in particular L 1 techniques.

Section 6 is devoted to magnetic Schrödinger semigroups; (besides its mathematical and physical interest, this class of semigroups illustrates sig-ni…cantly the fact that, in the above general theory, the positivity assumption on the unperturbed semigroup could be relaxed by using domination arguments; in principle, this strategy could even be used in full generality by exploiting the existence of a modulus of the unperturbed semigroup i.e. a minimal dominating positive contraction semigroup [START_REF] Kipnis | Majoration des semigroupes de contraction de L 1 et applications[END_REF]). According to a classical result, see e.g. [START_REF] Avron | Schrödinger operators with magnetic …elds. I General interactions[END_REF], the spectrum of the magnetic Schrödinger operator in L 2 is discrete if this is the case for the Schrödinger operator without magnetic potential; the pointwise diamagnetic inequality being the key point. Thus, the L 2 compactness results for Schrödinger operators are automatically translated into L 2 compactness results for magnetic Schrödinger operators. By following our L 1 approach and using the diamagnetic inequality, we give a compactness result in L 1 setting when the sublevel sets M are "thin at in…nity". This L 1 point of view complements the known hilbertian results on discreteness of magnetic Schrödinger operators. On the other hand, as far as we know, the situation where the sublevel sets M are not "thin at in…nity" has not been dealt with yet; we show here the existence of a spectral gap for magnetic Schrödinger operators under a condition involving the heat kernel and the sublevel sets of the potential, (see Theorem 45).

In Section 7, we deal with some aspects of weighted Laplacians, (see e.g. [START_REF] Davies | Heat Kernels and Spectral Theory[END_REF][31] [START_REF] Hel¤er | Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians[END_REF] for the interest of the subject); in particular, we revisit some problems which were considered in [START_REF] Hel¤er | Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians[END_REF] in connection with Fokker-Planck operators. We consider the weighted Laplacian

4 := 1 h 2 div(h 2 r) = 4 + 2 rh:r h which is (minus) the self-adjoint operator in L 2 (R N ; (dx)) associated to the Dirichlet form R R N jr'j 2 (dx) where (dx) = h 2 (x)dx with h 2 C 2 (R N
) and h(x) > 0 8x 2 R N : (We have restricted ourselves to 4 for simplicity but more general elliptic operators with smooth coe¢ cients can be dealt with similarly.) This operator is unitarily equivalent to the Schrödinger operator ) where is a real

4 4h h on L 2 (R N ; dx): In particular, if h(x) = e 2 (x
C 2 function on R N then 4h h = 1 4 jr (x)j 2 1 2 4 (x)
. Then, assuming that 1 4 jr (x)j 2 1 2 4 (x) is bounded from below, we can exploit our previous L 1 results on Schrödinger operators to give new (compactness) results on the subject. Thus, we consider the (non-convex) potential

(x) = 1 h N X j=1 ( 12
x 4 j + 2 x 2 j ) + 1 h I 2 N X j=1 jx j x j+1 j 2
with the convention x N +1 = x 1 where h > 0; > 0; < 0; I > 0 (which appears e.g. in [36][43]) and show that 4

( 1 4 jr (x)j 2 1 2 4 (x)) generates a (holomorphic) compact semigroup in L 1 (R N ; dx):
When is a uniformly strictly convex potential then by a classical result of D. Bakry and M. Emery (see e.g. [START_REF] Royer | Une initiation aux inégalités de Sobolev logarithmiques[END_REF] Théorème 3.1.29, p. 50) a logarithmic Sobolev inequality holds implying in particular the spectral gap (or Poincaré) inequality; we complement this result by showing that actually

4 ( 1 4 jr (x)j 2 1 2 4 (x)) generates a (holomorphic) compact semigroup in L 1 (R N ; dx):
We consider also the case of (nonpositive) polynomial potential

(x) = X j j C c x 2 1 1 x 2 2 2 :::x 2 N N ; (c > 0)
where i > 0 8i for at least one multi-index ; it is known (see [START_REF] Hel¤er | Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians[END_REF] Theorem 11.10 (ii), p. 120) that 4

( 1 4 jr (x)j 2 1 2 4 (x)) is resolvent compact in L 2 (R N ; dx); we show here that actually 4 ( 1 4 jr (x)j 2 1 2 4 (x)) generates a (holomorphic) compact semigroup in L 1 (R N ; dx).
The homogeneous (nonnegative) case

(x) = X j j=r c x 2 1 1 x 2 2 2 :::x 2 N N ; (c > 0)
is also dealt with but only in the simplest "elliptic"case r (x) 6 = 0 8x 6 = 0, (we refer to [START_REF] Hel¤er | Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians[END_REF] for a systematic analysis of polynomial potentials in L 2 setting). We deal also with spectral gaps when 1 4 jr (x)j 2 1 2 4 (x) is bounded from below; we give a su¢ cient condition for the existence of a spectral gap for 4 in L 2 (R N ; (dx)) under kernel estimates involving the sublevel sets of 1 4 jr (x)j 2 1 2 4 (x). In particular, if 1 4 jr (x)j 2 1 2 4 (x) > 0 and e (x) 2 L 1 (R N ; dx); the existence of a spectral gap for 4 is garanteed under the condition

sup M >0 lim C!+1 sup y2R N Z fx2 M ; jxj>Cg 1 (4 t) n 2 exp( jx yj 2 4t )dx < 1:
Thus, this condition provides us with a su¢ cient criterion in terms of sublevel sets of 1 4 jr (x)j 2 1 2 4 (x) for a probability measure e (x) dx on R N to satisfy the Poincaré inequality.

In Section 8, we deal with Witten Laplacians, i.e. weighted Hodge Laplacians, on 1-forms associated to the Witten Complex (i.e. the exterior di¤erential d of the De Rham Complex is replaced by d := e 2 (x) de 2 (x) where is a suitable smooth function; see e.g. [START_REF] Sjöstrand | Correlation asymptotics and Witten Laplacians[END_REF][43] and [START_REF] Hel¤er | Semiclassical Analysis, Witten Laplacians and Statistical Mechanics[END_REF] Chapter 2). The Witten Laplacian on 0-forms is unitarily equivalent to

4 (0) = 4 (0) + 1 4 jr j 2 1 2 4
(where 4 (0) = 4) while the Witten Laplacian on 1-forms is unitarily equivalent to 4

(1) = 4

(0)

Id + Hess where 1-forms are identi…ed with vector functions; both Laplacians are nonnegative and the spectral bottom of 4 (0) is zero when e (x) dx is a probability measure. The interest of Witten Laplacians in Statistical Mechanics stems in particular from the beautiful Hel¤er-Sjöstrand's covariance formula

Z (f (x) hf i)(g(x) hgi)e (x) dx = Z (4 (1) ) 1 df; dg e (x) dx;
where hf i = R f (x)e (x) dx (see [START_REF] Sjöstrand | Correlation asymptotics and Witten Laplacians[END_REF][43] and [START_REF] Hel¤er | Semiclassical Analysis, Witten Laplacians and Statistical Mechanics[END_REF] Chapter 2). The invertibility of 4 (1) is of course a key point; actually, it su¢ ces that the restriction of 4 (1) to exact 1-forms be invertible; (see [START_REF] Johnsen | On the spectral properties of Witten-Laplacians, their ranges projections and Brascamp-Lieb's inequality[END_REF] for the details). By combining L 1 results and hilbertian tools (Glazman's Lemma) we show here that if is convex (no strict convexity is needed) then the essential lower bound of 4 (0) is less than or equal to that of 4 (1) ; in particular 4 (1) is resolvent compact if 4 (0) is. We show also, for convex , that if 4 (0) has spectral gap and if Hess is not degenerate, i.e. its lowest eigenvalue is not identically zero, then the spectral bottom of 4 (1) is strictly bigger than that of 4

(and consequently 4 (1) is invertible if e (x) dx is a probability measure).

Regardless of any convexity assumption, we show also that if is the lowest eigenvalue of Hess , if 1 4 jr j 2 1 2 4 + is lower bounded and if 1) is also resolvent compact; in particular 4 (1) is resolvent compact if the sublevel sets of 1 4 jr j 2 1 2 4 + are "thin at in…nity". When such sublevel sets are not "thin at in…nity", we show the existence of a spectral gap for 4 (1) in terms of the heat kernel and these sublevel sets, (see Theorem 64).

4 (0) + is resolvent compact then 4 (
In the last section (Section 9) we come back to the general theory and deal with operators "T (V + V )" with inde…nite potentials V = V + V and consider them as perturbed operators

T V + + V provided that V is T V + -bounded.
The tools behind their treatment are di¤erent from the ones above used for T V + : According to [START_REF] Desch | Perturbations of positive semigroups in AL-spaces[END_REF] (see e.g. [START_REF] Banasiak | Perturbations of Positive Semigroups with Applications[END_REF] Chapter 5),

T V + + V : D(T V + ) ! L 1 ( ; )
generates a positive semigroup fW (t); t > 0g if and only if

lim !+1 r V ( T V + ) 1 < 1:
We show that if e tT ; t > 0 is holomorphic then so is fW (t); t > 0g : We show also that if e tT ; t > 0 is norm continuous then so is fW (t); t > 0g provided that

sup y2 Z fV >jg G V + (x; y)V (x) (dx) ! 0 as j ! +1
where G V + (x; y) is the kernel of (1 T V + ) 1 : In both cases, fW (t); t > 0g is shown to be a compact semigroup under (8) (where M are the sublevel sets of V + ). We study also stability of essential types (or essential spectral bounds). Indeed, we assume that is a locally compact metric space endowed with a locally …nite Borel measure > 0 and that = [ n n where n is a nondecreasing sequence of compact subsets. We show that n e t(T V + +V ) ; t > 0 o and n e tT V + ; t > 0 o have the same essential spectrum and consequently the same essential type provided that

lim n!1 sup y2 Z c n V (x)G V + (x; y) (dx) = 0;
this stability of essential type implies that n e t(T

adjoint in L 2 ( ). We note that V is not a priori T pV + -bounded in L p ( );
it is shown in [START_REF] Mokhtar-Kharroubi | New form-bound estimates for many-particle Schrödinger-type Hamiltonians[END_REF] that in L 2 ( ), V is form-bounded with respect to T 2V + with relative form-bound less than or equal to lim !+1 r V ( T V + ) 1 and (minus) A 2 turns out to be a form-sum

A 2 = ( T 2V + ) u ( V ).
Finally, we show by interpolation arguments how e tAp ; t > 0 inherites from n e t(T V + +V ) ; t > 0 o various compactness results and spectral stability results.

An abridged version of this paper will be published soon.

2 Compactness properties of sub-stochastic semigroups in L 1 ( ; )

In all this section ( ; A; ) denotes a measure space and fU (t); t > 0g is a positive c 0 -semigroup of contractions on L 1 ( ; ) (i.e. a sub-stochastic c 0 -semigroup) with generator T:

Let V : ( ; ) ! [0; +1]
be a measurable function. (Inde…nite potentials will be dealt with in Section 9.) Let V n := V ^n and e t(T Vn) ; t > 0 be the c 0 -semigroup generated by

T V n . It is elementary to see that e t(T V n+1 ) f e t(T Vn) f 8f 2 L 1 + ( ; ) so that U V (t)f := lim n!+1
e t(T Vn) f de…nes a semigroup. The semigroup fU V (t); t > 0g is a priori strongly continuous for t > 0 only. We say that V is admissible for fU (t); t > 0g if fU V (t); t > 0g is a c 0 -semigroup, i.e. is strongly continuous at the origin. In such a case, T V , the generator of fU

V (t); t > 0g, is an extension of T V : D(T ) \ D(V ) ! L 1 ( ; ): A su¢ cient condition of admissibility is that D(T ) \ D(V ) be dense in L 1 ( ; ) ([ 83
] Proposition 2.9). Actually the above considerations hold also in all L p spaces. On the other hand, the following known result is peculiar to L 1 -setting [66][83]; for reader's convenience, we recall brie ‡y its proof (as given in [START_REF] Voigt | Absorption semigroups, their generators and Schrödinger semigroups[END_REF] Lemma 4.1) in a slightly di¤erent form.

Lemma 2 Let V > 0 be admissible for fU (t); t > 0g. Then D(T V ) D(V ) and V is T V -bounded.

Proof: For a bounded potential W and f 2 D(T

) \ L 1 + ( ; ) we have d dt e t U W (t)f = d dt Z e t U W (t)f d = Z d dt h e t U W (t)f i d = Z (T W ) h e t U W (t)f i d = Z (T ) h e t U W (t)f i d Z W h e t U W (t)f i d e t kW U W (t)f k and consequently Z +1 0 e t kW U W (t)f k dt Z +1 0 d dt e t U W (t)f dt = kf k : Thus Z +1 0 e t kV n U Vm (t)f k dt kf k ; 8m > n since U Vm (t) U Vn (t)
e t kV U V (t)f k dt kf k which is nothing but V ( T V ) 1 f kf k for f 2 D(T ) \ L 1 + ( ; ). Finally the density of D(T ) \ L 1 + ( ; ) in L 1 + ( ; ) and the fact that L 1 ( ; ) = L 1 + ( ; ) L 1 + ( ; ) show that V ( T V ) 1 is a bounded operator or equiv- alently V is T V -bounded.
It seems that we cannot hope a priori (see Remark 5 below) that U V (t) maps continuously L 1 ( ; ) into D(V ) for t > 0: We show however a crucial "weak type" estimate under a suitable assumption on the perturbed semigroup fU V (t); t > 0g itself.

Lemma 3 We assume that L 1 ( ; ) is separable and that fU V (t); t > 0g is such that 8t > 0; sup "2]0;1] U V (t + ")1 U V (t)1 " L 1 ( ; ) < +1 (12)
where U V (t) is the dual operator of U V (t). Then, for almost all t > 0; there exists a positive constant c t such that

Z fV >M g (U V (t)f ) (dx) c t kf k M ; 8 f 2 L 1 + ( ; ); 8 M > 0: (13)
In particular, (12) is satis…ed if

8f 2 L 1 ( ; ); ]0; +1[ 3 t ! Z U V (t)f is di¤ erentiable (14) or if ]0; +1[ 3 t ! U V (t) 2 L(L 1 ( ; )) is locally lipschitz.
Proof: As in the proof of Lemma 2, we have

d dt kU Vn (t)f k kV n U Vn (t)f k so that, for any 0 a < b < +1; Z b a kV n U Vn (s)f k ds kU Vn (a)f k kU Vn (b)f k and, for n > m; Z b a kV m U Vn (s)f k ds kU Vn (a)f k kU Vn (b)f k : Letting n ! +1 gives Z b a kV m U V (s)f k ds kU V (a)f k kU V (b)f k so that letting m ! +1 we get by monotone convergence theorem Z b a kV U V (s)f k ds kU V (a)f k kU V (b)f k
showing in particular that U V (s)f 2 D(V ) for almost all s. We note that s ! kV U V (s)f k is lower semicontinuous (and thus measurable) as an increasing limit of continuous functions and is locally Lebesgue integrable. Hence, for any f 2 L 1 ( ; ),

[0; +1[ 3 t ! Z t 0 kV U V (s)f k ds is di¤erentiable a.e.
(on the set E f of Lebesgue points of s ! kV U V (s)f k) with derivative kV U V (t)f k, see e.g. [START_REF] Schwartz | Analyse IV. Applications de la théorie de la mesure[END_REF] Théorème 6.1.23, p. 30. Taking

f 2 L 1 + ( ; ), t 2 E f 1 " Z t+" t kV U V (s)f k ds 1 " [ kU V (t)f k kU V (t + ")f k] = Z U V (t)1 U V (t + ")1 " f c t kf k where c t := sup "2]0;1] " 1 kU V (t + ")1 U V (t)1k L 1 ( ; ) ; so that letting " ! 0 we get kV U V (t)f k c t kf k (t 2 E f ): If L 1 ( ; ) is separable then so is L 1 + ( ; ). Let D L 1 + ( ; ) be a denu- merable set dense in L 1 + ( ; ) and let E := \ f 2D E f : Finally kV U V (t)f k c t kf k ; f 2 D; t 2 E where the complement set (in R + ) E c = [ f 2D E c f has zero Lebesgue measure. It follows that for all f 2 D M Z fV >M g U V (t)f Z fV >M g V U V (t)f kV U V (t)f k c t kf k and then Z fV >M g (U V (t)f ) (dx) c t kf k M ; f 2 L 1 + ( ; ) since D is dense in L 1 + ( ; ) and U V (t) is a bounded operator on L 1 ( ; ). The di¤erentiability of ]0; +1[ 3 t ! R U V (t)f for all f 2 L 1 ( ; ) amounts to 8t > 0; lim "!0 U V (t + ")1 U V (t)1
" exists in the weak star topology of L 1 ( ; ) which in turn implies the boundedness of " 1 kU V (t + ")1 U V (t)1k L 1 ( ; ) for " 2 ]0; 1] by the uniform boundedness principle. Finally

kU V (t + ")1 U V (t)1k L 1 ( ; ) kU V (t + ") U V (t)k L(L 1 ( ; )) = kU V (t + ") U V (t)k L(L 1 ( ; ))
show the last claim.

Remark 4 (i) The separability of L 1 ( ; ) holds e.g. if is a -…nite regular Borel measure on a separable metric space .

(ii) We note that the condition that ]0; +1[ 3 t ! U V (t) 2 L(L 1 ( ; )) be locally lipschitz is weaker than a di¤ erentiability condition on the perturbed semigroup fU V (t); t > 0g because the di¤ erentiability of a bounded semigroup fS(t); t > 0g in a Banach space X is equivalent to global Lipschitz conditions 8" > 0; 9C " > 0; kS(t) S(s)k L(X) C " jt sj ; 8t; s > "; see e.g. [START_REF] Iley | Perturbations of di¤erentiable semigroups[END_REF] Lemma 2.1. Note that (14) is also much weaker than a di¤ erentiability condition of the perturbed semigroup fU V (t); t > 0g. For instance, if we consider the translation semigroup

U (t)f = f (x t) in L 1 (R; dx) then (14) is satis…ed by the (non di¤ erentiable) perturbed semigroup U V (t)f = e R x x t V (s)ds f (x t) provided that V is di¤ erentiable and 8t > 0; y ! V 0 (y + t)e R y+t y V (s)ds V 2 (y + t)e R y+t y V (s)ds is bounded.
Actually, in this example, fU V (t); t > 0g is not even norm continuous; this shows that a priori there is no connection between Conditions (12) or [START_REF] Carlen | Logarithmic Sobolev Inequalities and Spectral Gaps. Recent advances in the theory and applications of mass transport[END_REF] and norm continuity of the perturbed semigroup. Such conditions deserve to be investigated more deeply for their own sake. In particular, su¢ cient conditions in terms of fU (t); t > 0g and V would be very important; note that fU V (t); t > 0g is holomorphic if fU (t); t > 0g is [START_REF] Arendt | Absorption semigroups and Dirichlet boundary conditions[END_REF] [START_REF] Ishikawa | Analyticity of absorption semigroups[END_REF]. A natural condition to be investigated is of course the di¤ erentiability of the perturbed semigroup fU V (t); t > 0g; we mention that in general the di¤ erentiability property of a semigroup is not stable by bounded perturbations [START_REF] Renardy | On the Stability of Di¤erentiability of Semigroups[END_REF], see however [START_REF] Doytchinov | On Perturbations of Di¤erentiable Semigroups[END_REF] [START_REF] Iley | Perturbations of di¤erentiable semigroups[END_REF] for the positive results in this direction; the case of unbounded perturbations seems to be open.

(iii) We note …nally that if U V (t) is an integral operator with kernel

p V t (x; y) then U V (t)1 = R p V t (x; :) (dx) and (12) is satis…ed if ]0; +1[ 3 t ! Z p V t (x; :) (dx) 2 L 1 ( ; ) is locally lipschitz. Remark 5 Despite the denseness of D in L 1 + ( ; ), it is unclear whether the estimate kV U V (t)f k c t kf k ; f 2 D; t 2 E implies that V U V (t)
is a bounded operator on L 1 ( ; ) for t 2 E which is a stronger conclusion than (13):

We give now: Lemma 6 Let fV (t); t > 0g be a c 0 -semigroup on L 1 ( ; ) with generator G: If the resolvent ( G) 1 is a weakly compact operator for some (or equivalently all) 2 (G) then it is a compact operator for all 2 (G):

Proof: The resolvent identity ( G) 1 ( G) 1 = ( )( G) 1 ( G) 1 ; ; 2 (G)
shows that the weak compactness of ( G) 1 implies the the weak compactness of ( G) 1 : By the classical Dunford-Pettis'theorem (see e.g. [2] Corollary 5.88, p. 344) the product of two weakly compact operators on L 1 ( ; ) is a compact operator so that

( G) 1 ( )( G) 1 ( G) 1 = ( G) 1 ! 0 as ! +1
shows that ( G) 1 is a compact operator. We note that if ( T ) 1 is weakly compact (or equivalently compact) then ( T V ) 1 is also weakly compact by domination. Similarly if U (t) is compact for t > 0 then so is U V (t): Thus, in all this section, it is understood that ( T ) 1 and U (t) are not (weakly) compact in L 1 ( ; ): We are now ready to show: Theorem 7 Let fU (t); t > 0g be a sub-stochastic c 0 -semigroup on L 1 ( ; ) with generator T and let V : ( ; ) ! [0; +1] be admissible for fU (t); t > 0g. Then T V is resolvent compact if and only if for all M > 0

( T V ) 1 : L 1 ( ; ) ! L 1 ( M ; ) is weakly compact. ( 15
)
A su¢ cient condition for [START_REF] Christ | On the @ equation in weighted L 2 norms in C 1[END_REF] to hold is that

( T ) 1 : L 1 ( ; ) ! L 1 ( M ; ) is weakly compact. ( 16
)
Proof: According to Lemma 6, it su¢ ces to show that T V is resolvent weakly compact. Let s(T V ) be the spectral bound of

T V : Let f = ( T V ) 1 g with > s(T V ) (g 2 B) where B is the unit ball of L 1 ( ; ). Since D(T V ) D(V ) and V is T V -bounded (Lemma 2) then there exists a constant c > 0 such that kV f k c kgk so that M Z fV (x)>M g jf (x)j (dx) Z fV (x)>M g V (x) jf (x)j (dx) Z V (x) jf (x)j (dx) c; 8g 2 B so that R fV (x)>M g jf (x)j (dx) ! 0 as M ! +1 uniformly in g 2 B. Thus we have decomposed f = ( T V ) 1 g as f 1 M + f 1 c M where f 1 c
M can be made as small in L 1 -norm as we want (uniformly in g 2 B) and f 1 M is a relatively weakly compact set by [START_REF] Christ | On the @ equation in weighted L 2 norms in C 1[END_REF]: This shows the …rst claim. Finally, the domination ( T V ) 1 ( T ) 1 shows that (16) implies (15): Under an additional technical assumption (which is satis…ed e.g. in denumerable state spaces, see Remark 9), we can show that (15) and ( 16

f ' M ' M T f + ' M V f = ' M h + ' M V f and k := f ' M satis…es the equation k T k + V k = ' M h + ' M V f + ' M T f T k:
We note that by assumption

' M h + ' M V f + ' M T f T k lives in a bounded subset of L 1 ( ; ). The set n f ' M ; khk L 1 ( ; ) 1 o is relatively compact in L 1 ( ;
) by the compactness of ( T V ) 1 and …nally

n f ; khk L 1 ( ; ) 1 o is relatively compact in L 1 ( M ; ) since ' M is equal to one on M , i.e. ( T ) 1 : L 1 ( ; ) ! L 1 ( M ; ) is compact.
Remark 9 Let L 1 (N; ) endowed with the counting measure . Consider an in…nite matrix fa i;j ; i; j 2 Ng such that a i;i 0, a i;j > 0 for i 6 = j and P i a i;j = 0: Let T 0 be the multiplication operator by fa i;i g i2N with domain

fu i g i 2 l 1 (N); fa i;i u i g i 2 l 1 (N)
and b A : D(T 0 ) ! l 1 (N) with ( b Au) i = P j6 =i a i;j u j : Note that P i6 =j a i;j = a j;j : If lim !+1 sup j P i6 =j a i;j a j;j < 1 then T := T 0 + b A : D(T 0 ) ! l 1 (N) generates a stochastic (i.e. mass preserving on the positive cone) c 0semigroup fU (t); t > 0g on l 1 (N) (see e.g. [START_REF] Kato | On the semi-groups generated by Kolmogoro¤'s di¤erential equations[END_REF]). Then D(T ) = D(T 0 ) is obviously invariant under the multiplication by any sequence fz i g i 2 l 1 (N). Consider now a nonnegative V = fV i g i . For any f M > M , the bounded sequence ' = f' i g i2N de…ned by ' i = 1 if V i f M and ' i = 0 otherwise, satis…es the conditions on ' M in Theorem 8.

We show now how to reach the conclusions of Theorem 7 (and also more stronger ones) by di¤erent means.

Theorem 10 Let fU (t); t > 0g be a sub-stochastic c 0 -semigroup on L 1 ( ; ) with generator T: Let V : ( ; ) ! [0; +1] be admissible for fU (t); t > 0g. We assume that for M > 0 and t > 0

U (t) : L 1 ( ; ) ! L 1 ( M ; ) is weakly compact. ( 17
)
Then:

(i) T V is resolvent compact.

(ii) If moreover the conditions (12) or ( 14) are satis…ed then fU V (t); t > 0g is a compact semigroup.

Proof: (i) Let P M : L 1 ( ; ) ! L 1 ( M ; ) be the restriction operator. Note that

P M ( T ) 1 = P M Z +1 0 e t U (t)dt = lim "!0 P M Z " 1 " e t U (t)dt
where the convergence holds in operator norm. Let us show [START_REF] Davies | Quantum dynamical semigroups and the neutron di¤usion equation[END_REF] or equivalently that P M ( T ) 1 is weakly compact. It su¢ ces to show that

P M Z " 1 " e t U (t)dt = Z " 1 " e t P M U (t)dt
is a weakly compact operator. This is a strong integral (not a Bochner integral) of a bounded, strongly continuous W (L 1 ( ; ); L 1 ( M ; ))-valued mapping where W (L 1 ( ; ); L 1 ( M ; )) is the Banach space of weakly compact operators from L 1 ( ; ) into L 1 ( M ; ). By [START_REF] Schuchtermann | On weakly compact operators[END_REF] or [START_REF] Mokhtar-Kharroubi | On the strong convex compactness property for the strong operator topology and related topics[END_REF] R " 1 " e t P M U (t)dt is a weakly compact operator. Then the …rst claim is a consequence of Theorem 7.

(ii) We can choose t as small as we want such that (13) is satis…ed with t = t. Let f = U V (t)g with g 2 B the unit ball of L 1 ( ; ): We note that (13) implies that R fV (x)>M g jf (x)j (dx) ! 0 as M ! +1 uniformly in g 2 B: On the other hand

jf j = U V (t)g U V (t) jgj U (t) jgj
so that, by [START_REF] Davies | L 1 -Properties of Intrinsic Schrödinger semigroups[END_REF], the restriction to M of U V (t)g; g 2 B is relatively weakly compact by domination and then, by arguing as in the proof of Theorem 7, one sees that U V (t)g; g 2 B is a relatively weakly compact subset of L 1 ( ; ), i.e. U V (t) is a weakly compact operator for all t > t and consequently for all t > 0:

Actually, U V (t) is a compact operator for all t > 0 since U V (t) = U V ( t 2 )U V ( t 2
) and the product of two weakly compact operators on L 1 ( ; ) is a compact operator (see e.g. [START_REF] Aliprantis | Positive Operators[END_REF] Corollary 5.88, p.

344):

We give now an alternative proof of Theorem 10(ii).

Theorem 11 Let (16) be satis…ed and let fU V (t); t > 0g be norm continuous. Then U V (t) is compact for all t > 0.

Proof: It is a standard fact from semigroup theory (see e.g. [START_REF] Nagel | One-Parameter Semigroups of Positive Operators[END_REF] Theorem 1.25, p 41) that under the norm continuity of fU V (t); t > 0g, the compactness of ( T V ) 1 is equivalent to the compactness of U V (t) for t > 0. We can give here a di¤erent argument adapted to our particular context. We have (for large and) for any t > 0 and " > 0

( T V ) 1 = Z +1 0 e t U V (t)dt > Z t+" t e t U V (t)dt
so that, for any " > 0; " 1 R t+" t e t U V (t)dt is a weakly compact operator by domination. Letting " ! 0 and using the right continuity of t ! U V (t) in operator norm we obtain that U V (t) is a weakly compact operator for all t > 0 and consequently, as previously, U V (t) is compact for all t > 0.

One sees then how important is the norm continuity of fU V (t); t > 0g; it provides us with a useful mean to translate compactness properties from the resolvent ( T V ) 1 to the semigroup U V (t). It is an interesting open problem to decide whether the norm continuity of fU (t); t > 0g implies that of fU V (t); t > 0g. This problem is not covered by the paper [START_REF] Mátrai | On perturbations preserving the immediate norm continuity of semigroups[END_REF] dealing with unbounded perturbations preserving immediate norm continuity of the semigroup. We provide here a solution to this open problem.

Theorem 12 Let V : ( ; ) ! [0; +1[ and V n := V ^n:

(i) Then for all …nite C > 0

sup t C e t(T Vn) f U V (t)f e C [V V n ] (1 T V ) 1 f ; 8f 2 L 1 + ( ; ): In particular, if [V V n ] (1 T V ) 1 L(L 1 ( ; )) ! 0 as n ! +1 and if fU (t); t > 0g is norm continuous then fU V (t); t > 0g is also norm continu- ous.
(ii) In particular, let (1 T V ) 1 be an integral operator with kernel G V (x; y): If fU (t); t > 0g is norm continuous and if

sup y2 Z fV >ng G V (x; y)V (x) (dx) ! 0 as n ! +1
then fU V (t); t > 0g is also norm continuous.

Proof : Note …rst that both V and V n are T V -bounded so that the sequence

[V V n ] (1 T V ) 1
n of bounded operators converges strongly to zero. According to the general theory e t(T Vn) f ! U V (t)f for all f 2 L 1 ( ; ) uniformly in t 2 [0; C]. We start with the Duhamel formula (for a positive bounded perturbation) and f 2 L 1 + ( ; )

e t(T Vn) f = e t(T V n+k ) f + Z t 0 e (t s)(T V n+k ) [V n+k V n ] e s(T V n+k ) f ds: By letting k ! +1; V n+k (x) V n (x)!V (x) V n (x)
a.e. and then

e t(T Vn) f = U V (t)f + Z t 0 U V (t s) [V V n ] U V (s)f ds:
The additivity of the norm on the positive cone shows that

e t(T Vn) f U V (t)f = Z t 0 U V (t s) [V V n ] U V (s)f ds = Z t 0 kU V (t s) [V V n ] U V (s)f k ds Z t 0 k[V V n ] U V (s)f k ds = Z t 0 [V V n ] U V (s)f ds = [V V n ] Z t 0 U V (s)f ds [V V n ] Z C 0 U V (s)f ds e C [V V n ] Z C 0 e s U V (s)f ds for all t C where C > 0 is arbitrary. Hence sup t C e t(T Vn) f U V (t)f e C [V V n ] (1 T V ) 1 f ; 8f 2 L 1 + ( ; ) and sup t C e t(T Vn) U V (t) e C [V V n ] (1 T V ) 1 :
Finally, if fU (t); t > 0g is norm continuous then so is e t(T Vn) ; t > 0 because V n is a bounded perturbation [START_REF] Phillips | Perturbation Theory for Semi-Groups of Linear Operators[END_REF] so that the last operator norm estimate ends the proof of (i). If (1 T V ) 1 is an integral operator with kernel G V (x; y) then an elementary calculation shows that

[V V n ] (1 T V ) 1 L(L 1 ( )) = sup y2 Z fV >ng G V (x; y)V (x) (dx)
and this combined with (i) end the proof of (ii). The category of holomorphic semigroups is also particularly interesting since the holomorphy of fU (t); t > 0g implies that of fU V (t); t > 0g [4] [START_REF] Ishikawa | Analyticity of absorption semigroups[END_REF] and therefore Theorem 11 implies:

Corollary 13 Let (16) be satis…ed. If fU (t); t > 0g is holomorphic or under the conditions of Theorem 12 (ii), fU V (t); t > 0g is a compact semigroup.

Remark 14 It is not di¢ cult to see that ( T V ) 1 is compact if and only if R t 0 U V (s)
ds is for all t > 0 (the argument holds for general c 0 -semigroups in Banach spaces). Thus R t 0 U V (s)ds is a compact operator on L 1 ( ; ) under Assumption (16) only.

Remark 15 (i)

The assumption that ( T ) 1 (resp. U (t)) : L 1 ( ; ) ! L 1 ( M ; ) is weakly compact is trivially satis…ed if the sublevels sets have …nite measure and ( T ) 1 (resp. U (t)) maps continuously L 1 ( ; ) into L p ( ; ) for some p > 1 for any Borel set with …nite measure, (e.g. for ultracontractive symmetric Markov semigroups); this follows from the fact that for p > 1; a bounded subset of L p ( ; ) is equi-integrable.

(ii) When L 1 ( ; ) is separable, such weak compactness assumptions imply that ( T ) 1 : L 1 ( ; ) ! L 1 ( M ; ) (resp. U (t) : L 1 ( ; ) ! L 1 ( M ; )) is an integral operator with a measurable kernel (see the remark in [START_REF] Dunford | Linear Operators, Part 1: General Theory[END_REF] p. 508) and this clearly implies that ( T ) 1 : L 1 ( ; ) ! L 1 ( ; ) (resp. U (t) : L 1 ( ; ) ! L 1 ( ; )) is an integral operator with a measurable kernel (provided that [ M M = , i.e. V is …nite a.e). In particular, this is the case of ultracontractive symmetric Markov semigroups (see also [START_REF] Simon | Schrödinger semigroups[END_REF] Corollary A.1.2).

In the special case where fU (t); t > 0g operates on all L p spaces, we can extend the above results to L p spaces by interpolation. Suppose that fU p (t); t > 0g are c 0 -semigroups on L p ( ; ) with generator T p (p > 1) such that U p (t) and U q (t) coincide on L p ( ; ) \ L q ( ; ): We note fU (t); t > 0g instead of fU 1 (t); t > 0g. Then, as in the L 1 case, we de…ne the semigroups fU pV (t); t > 0g with generator T pV and fU pV (t); t > 0g are strongly continuous if and only if fU V (t); t > 0g is ([83] Proposition 3.1). (We point out that in T pV and U pV (t), the subscript pV is not the product of p and V !) Then using the compactness interpolation theorem for -…nite measures (see e.g. [START_REF] Davies | Heat Kernels and Spectral Theory[END_REF] Proof: By interpolation (see e.g. [START_REF] Davies | Heat Kernels and Spectral Theory[END_REF] Theorem 1.6.1, p. 35), the generator of the self-adjoint semigroup fU 2V (t); t > 0g is resolvent compact and then the semigroup itself is compact for t > 0: Actually this result can also be obtained as follows: being self-adjoint, ]0; +1[ 3 t ! U 2V (t) 2 L(L 2 ( ; )) is continuous in operator norm topology and then we can argue as in the proof of Theorem 11 by using compactness results by domination in L p spaces when p > 1 (Dodds-Fremlin's Theorem); see e.g. [START_REF] Aliprantis | Positive Operators[END_REF] Theorem 5.20, p. 286. The case p 6 = 2 (p > 1) follows by interpolation again.

Remark 18 Note that if fU 2 (t); t > 0g is self-adjoint then, under (16) only, the semigroup fU V (t); t > 0g is not a priori compact on L 1 ( ; ):

We end this section by showing that the basic assumption (17) is stable by subordination. We recall …rst some notions on subordinate semigroups. Let f 2 C 1 ((0; +1)) be a Bernstein function, i.e.

f > 0; ( 1) k d k f (x) dx k 0 8k 2 N:
It is characterized by the representation e tf (x) = R +1 0 e xs t (ds) (t > 0) where ( t ) t>0 is a convolution semigroup of measures on [0; +1) (see e.g. [START_REF] Jacob | Pseudo-Di¤ erential Operators & Markov Processes[END_REF] Theorem 3.9.7, p. 177). Let fU (t); t > 0g be a contraction semigroup. We can de…ne (see [START_REF] Jacob | Pseudo-Di¤ erential Operators & Markov Processes[END_REF] Chapter 4 for the details) the so-called subordinate semigroup U f (t); t > 0 acting as

' 2 L 1 (R N ) ! U f (t)' = Z +1 0 (U (s)') t (ds) 2 L 1 (R N ):
Theorem 19 Let fU (t); t > 0g be a positive contraction semigroup satisfying (17): Let f be a Bernstein function such that f (x) ! +1 as x ! +1: Then the subordinate semigroup U f (t); t > 0 satis…es also [START_REF] Davies | L 1 -Properties of Intrinsic Schrödinger semigroups[END_REF]:

Proof: Note …rst that f (x) ! +1 as x ! +1 (or equivalently: for all t > 0, e tf (x) ! 0 as x ! +1) amounts to t (f0g) = 0 8t > 0: This implies that

Z " 1 " U (s) t (ds) U f (t) t ([0; "[) + t ( " 1 ; +1 ) ! 0 as " ! 0; so that Z " 1 " P M U (s) t (ds) P M U f (t) ! 0 as " ! 0:
It su¢ ces then to show that R " 1 " P M U (s) t (ds) is a weakly compact operator. By assumption, 8s > 0, P M U (s) is a weakly compact operator. Moreover

s > 0 ! P M U (s) 2 L(L 1 (R N ); L 1 ( M ))
is strongly continuous and bounded. It follows from [START_REF] Schuchtermann | On weakly compact operators[END_REF] or [START_REF] Mokhtar-Kharroubi | On the strong convex compactness property for the strong operator topology and related topics[END_REF] that the strong integral R " 1 " P M U (s) t (ds) is a weakly compact operator.

Applications to convolution semigroups

This section is devoted to speci…c results on convolution semigroups in euclidean spaces because of their importance in applications. More general situations are dealt with in Section 4. Let R N be a Borel subset. We say that is "thin at in…nity" if j \ B(z; 1)j ! 0 as z ! 1 [START_REF] Davies | Heat Kernels and Spectral Theory[END_REF] where B(z; 1) is the ball with radius 1 centered at z 2 R N and jj refers to Lebesgue measure. We start with a basic result.

Lemma 20 Let H: ' 2 L 1 (R N ) ! R h(x y)'(y)dy 2 L 1 (R N ) be a con- volution operator with h 2 L 1 + (R N ). Then H : ' 2 L 1 (R N ) ! L 1 ( ) is compact if and only if sup y2R N Z \fjxj>cg h(x y)dx ! 0 as c ! +1: (19)
Moreover (19) is satis…ed if is "thin at in…nity".

Proof: We note …rst that the continuity of y 2 R N ! h y (:) 2 L 1 (R N ) (where h y (:) : x ! h(x y) is the translation of h(:) by a vector y) shows that H : L 1 (R N ) ! L 1 ( ) is compact for any bounded Borel set . On the other hand, if H :

' 2 L 1 (R N ) ! L 1 ( ) is compact then \fjxj>cg H L(L 1 (R N );L 1 ( )) ! 0 as c ! +1 (we
Indeed, let " > 0 be arbitrary and let D > 0 be such that Z h(x y)dx " for all jyj > D:

It
Since

y 2 R N ! h y (:) 2 L 1 (R N ) is continuous then fh y (:); jzj Dg is compact subset of L 1 (R N )
and consequently fh y (:); jzj Dg is an equi-integrable subset of L 1 (R N ) so that (21) is true. It su¢ ces now to show that (20) is satis…ed if is "thin at in…nity". We observe …rst that (18) is actually equivalent to

8R > 1; j \ B(y; R)j ! 0 as y ! 1 (22)
where B(y; R) is the ball with radius R centered at y 2 R N . It su¢ ces to observe that j \ B(y; R)j P J R i=1 j \ B(y i ; 1)j where we have covered B(y; R) by a …nite number J R (depending on R only) of balls B(y i ; 1) with radius 1. We write

Z h(x y)dx = Z y h(z)dz = Z ( y)\B(0;R) h(z)dz + Z ( y)\B(0;R) c h(z)dz Z ( y)\B(0;R) h(z)dz + Z B(0;R) c h(z)dz
where B(0; R) c is the exterior of the ball B(0; R). The invariance of Lebesgue measure by translation yields j( y) \ B(0; R)j = j \ B(y; R)j :

Finally, for any " > 0 we choose R large enough so that R B(0;R) c h(z)dz < " and then R ( y)\B(0;R) h(z)dz ! 0 as jyj ! +1 by ( 22) and (23). We consider now the convolution semigroups

U p (t) : f 2 L p (R N ) ! Z f (x y)m t (dy) 2 L p (R N )
de…ned in the Introduction. Such convolution semigroups, related to Lévy processes, cover many examples of practical interest such as Gaussian semigroups, -stable semigroups, relativistic Schrödinger semigroups, relativistic -stable semigroup etc. (see [START_REF] Jacob | Pseudo-Di¤ erential Operators & Markov Processes[END_REF] Chapter 3). The semigroups fU p (t); t > 0g t 0 are strongly continuous positive contractions on L p (R N ) for 1 p < +1 with generator T p . We recall that

( T ) 1 f = Z f (x y)m (dy) where c m ( ) = Z +1 0 e t c m t ( )dt = 1 + F ( ) :
We make the assumption that

9G 2 L 1 + (R N ) such that c G ( ) = 1 + F ( ) : (24)
Note that (24) is satis…ed if, for all t > 0, m t is a function, i.e. U (t) is a convolution operator with a kernel p t (:) 2 L 1 + (R N ): As a consequence of Lemma 20 we have: Theorem 21 Let (24) be satis…ed. If the sublevel sets M are "thin at in…nity" then T V is resolvent compact.

Since fU 2 (t); t > 0g is self-adjoint for real characteristic exponent then Theorem 17 implies: Corollary 22 We assume that the characteristic exponent is real. Let [START_REF] Dunford | Linear Operators, Part 1: General Theory[END_REF] be satis…ed and M be "thin at in…nity". Then fU pV (t); t > 0g are compact semigroups for all p > 1.

The fact that m is a function if e tF ( ) 2 L 1 (R N ) (t > 0) implies:

Corollary 23 We assume that e tF ( ) 2 L 1 (R N ) (t > 0): Then T V is resolvent compact if the sublevel sets M are "thin at in…nity".

Thus, the semigroups generated by (non-selfadjoint) second order elliptic operators with constant coe¢ cients, the -stable semigroup, the relativistic schrodinger operator or more generally the relativistic -stable semigroup are all covered by Corollary 23. These examples are also covered by the following theorem with a stronger conclusion.

Theorem 24 Let f be a Bernstein function such that f (x) ! +1 as x ! +1 and let U f (t); t > 0 be the corresponding subordinate Brown-

ian semigroup on L 1 (R N ). Then n U f V (t); t > 0 o
is a compact semigroup if the sublevel sets M are "thin at in…nity".

Proof: Let fU (t); t > 0g be the heat semigroup. Since (17) be clearly sat-is…ed, then Theorem 19 shows that (17) (and consequently [START_REF] Davies | Quantum dynamical semigroups and the neutron di¤usion equation[END_REF]) is satis…ed by its subordinate semigroup U f (t); t > 0 . The latter being holomorphic, Theorem 11 shows that

n U f V (t); t > 0 o is a compact semigroup.
Remark 25 We note that the geometric -stable semigroup corresponding to F ( ) = ln(1 + j j ) where 0 < 2 satis…es e tF ( ) 2 L 1 (R N ) for t > N only so that Corollary 23 does not apply a priori. However, by subordination to heat semigroup, this case is covered by Theorem 24.

Remark 26 (i) Note that we have now a stronger version of Theorem 1 since, under (3), 4 V generates a (holomorphic) compact semigroup in L 1 (R N ): In particular, this is the case in L 1 (R 2 ) for the potential (1). Indeed, in this case, M = n (x 1 ; x 2 ); jx 2 j M jx 1 j o : It su¢ ces to restrict ourselves to

+ M := M \ f(x 1 ; x 2 ); x 1 > 0; x 2 > 0g = (x 1 ; x 2 ); x 2 M x 1
and to consider for instance the case where we move the ball B(z; 1) (centered at z = (z 1 ; z 2 ) with z 1 > 0) by letting z 1 ! +1. The set B(z; 1)

\ + M is included in f(x 1 ; x 2 ); z 1 1 x 1 z 1 + 1g \ + M whose Lebesgue measure is equal to Z z 1 +1 z 1 1 M x 1 dx 1 = M ln( z 1 + 1 z 1 1
) ! 0 as z 1 ! +1:

(ii) Similar arguments apply to (x 2 y 2 + x 2 z 2 + y 2 z 2 ) in L 1 (R 3 ); see Section 7 for much more examples arising in the study of weighted Laplacians.

Remark 27

We note that we can easily deal with non translation-invariant positive semigroups fU (t); t > 0g on L 1 ( ; dx) where R N by using domination arguments provided that fU (t); t > 0g admits a kernel estimate of convolution type. More systematic results, including spectral gaps, are given in the next two sections in the more general context of L 1 spaces over metric measure spaces.

Metric measure spaces and compactness

The interest of Markov processes in metric spaces (see e.g. [START_REF] Grigor'yan | Heat kernels on metric measure spaces with regular volume growth[END_REF] and references therein) suggests naturally to investigate the compactness (or spectral gap) problems in Lebesgue spaces over metric measure spaces. Let ( ; d) be a metric space and be a Borel measure on which is …nite on bounded Borel sets. We assume in this section and in the next one that L 1 ( ; ) is separable. Let fU (t); t > 0g be a sub-stochastic c 0 -semigroup on L 1 ( ; ) with generator T . We show here how the existence of a metric allows us to precise further some results of Section 2.

Theorem 28 We assume that (1 T ) 1 : L 1 ( ; ) ! L 1 ( ) is weakly compact for any bounded Borel set . Let G 1 (x; y) be the kernel of (1

T ) 1 . If lim C!+1 sup y2 Z fx2 M ; d(x;x 0 )>Cg G 1 (x; y) (dx) = 0; 8M > 0 ( 25
)
(for some x 0 2 ) then T V is resolvent compact.

Proof: Note that (25) is x 0 -independent. As in Remark 15 (ii), the existence of the kernel G 1 (x; y) follows from the separability of L 1 ( ; ) and the weak compactness assumption, see the remark in [START_REF] Dunford | Linear Operators, Part 1: General Theory[END_REF] p. 508. One sees, by domination, that (1 T V ) 1 : L 1 ( ; ) ! L 1 ( ) is also weakly compact for any bounded Borel set and then (1

T V ) 1 has also a kernel G V 1 (x; y): We decompose (1 T V ) 1 as (1 T V ) 1 = c M (1 T V ) 1 + fx2 M ;d(x;x 0 )>Cg (1 T V ) 1 + fx2 M ;d(x;x 0 )<Cg (1 T V ) 1
where c M is the complement of the sublevel set M : Since

fx2 M ;d(x;x 0 )<Cg (1 T V ) 1 fx2 M ;d(x;x 0 )<Cg (1 T ) 1
then, by our assumption, fx2 M ;d(x;x 0 )<Cg (1 T V ) 1 is weakly compact. Moreover, we saw in the proof of Theorem 7 that the norm of c M (1 T V ) 1 goes to zero as M ! +1: The norm of fx2 M ;d(x;x 0 )>Cg (1 T V ) 1 is less than or equal to that of fx2 M ;d(x;x 0 )>Cg (1 T ) 1 i.e.

sup y2 Z fx2 M ; d(x;x 0 )>Cg G 1 (x; y) (dx): Thus (1 T ) 1 fx2 M ;d(x;x 0 )<Cg (1 T V ) 1 L(L 1 ( ; ))
is arbitrarily small for M and C large enough. Hence (1 T V ) 1 is weakly compact and Lemma 6 ends the proof. We note that under the conditions of Theorem 12 (ii) fU V (t); t > 0g is norm continuous and then Theorem 28 implies the compactness of the semigroup fU V (t); t > 0g (see Theorem 11). We can also derive this result di¤erently under other conditions.

Theorem 29 We assume that (12) or ( 14) is satis…ed. Let U (t) : L 1 ( ; ) ! L 1 ( ) be weakly compact for any bounded Borel set and t > 0 and let p t (x; y) be its kernel. If for all t > 0 lim

C!+1 sup y2 Z fx2 M ; d(x;x 0 )>Cg p t (x; y) (dx) = 0 (26)
(for some x 0 2 ) then fU V (t); t > 0g is a compact semigroup.

Proof: Note that (26) is x 0 -independent. Arguing as in the previous proof, one sees that U (t) and U V (t) have kernels p t (x; y) and p V t (x; y): We decompose U V (t) as

U V (t) = c M U V (t) + fx2 M ;d(x;x 0 )>Cg U V (t) + fx2 M ;d(x;x 0 )<Cg U V (t) (27)
where c M is the complement of the sublevel set M : Since

fx2 M ;d(x;x 0 )<Cg U V (t) fx2 M ;d(x;x 0 )<Cg U (t)
then, by our assumption, the third operator in ( 27) is weakly compact. Moreover, by the weak-type estimate (13) (a consequence of (12)), the norm of c M U V (t) goes to zero as M ! +1: Finally, the norm of fx2 M ;d(x;x 0 )>Cg U V (t) is less than or equal to that of fx2 M ;d(x;x 0 )>Cg U (t) i.e.

sup y2 Z fx2 M ; d(x;x 0 )>Cg p t (x; y) (dx): Thus U V (t) fx2 M ;d(x;x 0 )<Cg U V (t) L(L 1 ( ; ))
is arbitrarily small for M and C large enough. Hence U V (t) is weakly compact for all t > 0 and …nally fU

V (t); t > 0g is a compact semigroup since U V (t) = U V ( t 2 )U V ( t 2
) by Dunford-Pettis theorem.

We rely now Theorem 28 and Theorem 29 on the notion of sublevels sets "thin at in…nity". We introduce …rst: De…nition 30 We say that a Borel set is "thin at in…nity" if there exists a point y 2 such that for all M > 0 f \ B(y; M )g ! 0 as d(y; y) ! +1 [START_REF] Georgescu | Hamiltonian with purely discrete spectrum[END_REF] where B(y; M ) is the ball centered at y with radius M:

This de…nition is y-independent. We give now a basic preliminary result.

Lemma 31

We assume that

v(r) := sup x2 (B(x; r)) < +1; 8r > 0: (29)
Let H : ' 2 L 1 ( ; ) ! Z h(x; y)'(y) (dy)
with a kernel estimate of the form h(x; y) f (d(x; y)) where f : R + ! R + is nonincreasing and such that (for su¢ ciently large r) r ! f (r)v(r + 1) is nonincreasing and integrable at in…nity. Then: (i) H is bounded operator on L 1 ( ; ):

(ii) If a Borel set is "thin at in…nity" in the sense (28) then operator H : ' 2 L 1 ( ; ) ! L 1 ( ; ) is weakly compact.

Proof: (i) By domination, it su¢ ces to show that

' 2 L 1 ( ; ) ! Z f (d(x; y))'(y) (dy) 2 L 1 ( ; ) (30)
is a bounded operator. This holds if and only if there exists C > 0 such that Z f (d(x; y)) (dx) C 8y 2 :

We have

Z f (d(x; y)) (dx) = Z fd(x;y)<1g f (d(x; y)) (dx) + 1 X n=1 Z fn d(x;y)<n+1g f (d(x; y)) (dx) f (0) (B(y; 1)) + 1 X n=1 f (n) [ (B(y; n + 1)) (B(y; n))] = [f (0) f (1)] (B(y; 1)) + [f (1) f (2)] (B(y; 2)) + = 1 X n=0 [f (n) f (n + 1)] (B(y; n + 1)) (31)
which is …nite if

1 X n=0 f (n) (B(y; n + 1)) < 1; 1 X n=0 f (n + 1) (B(y; n + 1)) < 1 and then (31) is …nite if 1 X n=0 f (n)v(n + 1) < 1; 1 X n=0 f (n + 1)v(n + 1) < 1 or equivalently P 1 n=0 f (n)v(n+1) < 1 (since v(n) v(n+1)) which follows from the condition R +1 1 f (r)v(r + 1)dr < 1 because r ! f (r)v(r + 1) is nonincreasing. (ii)
f (d(x; y)) (dx) = 1 X n=0 Z fn d(x;y)<n+1g\ \fd(x;y)>cg f (d(x; y)) (dx) 1 X n=0 f (n) [fn d(x; y) < n + 1g \ \ fd(x; y) > cg] :
We note that

1 X n=m f (n) [fn d(x; y) < n + 1g \ \ fd(x; y) > cg] 1 X n=m f (n) [fn d(x; y) < n + 1g] = 1 X n=m f (n) [ (B(y; n + 1)) (B(y; n))] 1 X n=m f (n) [ (B(y; n + 1)) + (B(y; n))] c 1 X n=m f (n) [v(n + 1) + v(n)]
so that, for any " > 0 there exists an integer m such that

1 X n=m f (n) [fn d(x; y) < n + 1g \ \ fd(x; y) > cg] " uniformly in y 2 : It su¢ ces to show that m X n=0 f (n) [fn d(x; y) < n + 1g \ \ fd(x; y) > cg] ! 0 as c ! +1
uniformly in y 2 ; or equivalently for any n m Theorem 32 Let ( ; d; ) be a measure metric space. Let fU (t); t > 0g be a sub-stochastic c 0 -semigroup on L 1 ( ; ) with generator T and let (29) be satis…ed.

[fn d(x; y) < n + 1g \ \ fd(x; y) > cg] ! 0 as c ! +1 (32)
(i) We assume that (1 T ) 1 is an integral operator with a kernel G(x; y) satisfying an estimate of the form G(x; y) f (d(x; y)) where f : R + ! R + is nonincreasing and such that (for large r) r ! f (r)v(r+1) is nonincreasing and integrable at in…nity. If the sublevel sets M are "thin at in…nity" in the sense (28) then T V is resolvent compact.

(ii) Let L 1 ( ; ) be separable and let [START_REF] Benci | On a Discreteness Condition of the Spectrum of Schrödinger Operators with Unbounded Potential from Below[END_REF] or (14) be satis…ed. We assume that for each t > 0, U (t) is an integral operator with a kernel p t (:; :) satisfying an estimate of the form p t (:; :) f t (d(x; y)) where f t : R + ! R + is nonincreasing and such that (for large r) r ! f t (r)v(r + 1) is nonincreasing and integrable at in…nity. If the sublevel sets M are "thin at in…nity" in the sense (28) then fU V (t); t > 0g is a compact semigroup.

Note that under the conditions of Theorem 12 (ii) the conclusion of Theorem 32 (ii) follows also from Theorem 32 (i). Note also that if we consider the di¤erent examples of kernel estimates (5)(6) [START_REF] Auscher | Heat kernels and Analysis on Manifolds, Graphs and metric Spaces[END_REF] arising in the theory of Markov process

f t (r) := C t exp( r 2 Ct ); C t exp( r 1 C 1 t 1 ) or C t (1 + r t 1 ) ( + ) ;
one sees which volume growth r ! v(r) is compatible with each f t (:) in order to meet the conditions in Theorem 32. .

Metric measure spaces and spectral gaps

We recall …rst that the spectral bound s(A) := sup fRe ; 2 (A)g of the generator A of a positive semigroup on a Banach lattice X belongs to (A) (see e.g. [START_REF] Nagel | One-Parameter Semigroups of Positive Operators[END_REF] Theorem 1.1, p. 292) and this spectral bound coincides with the type of the semigroup when X is an L p ( ) space (see [START_REF] Weis | A short proof for the stability theorem for positive semigroups on L p[END_REF]). Moreover, a c 0 -semigroup fS(t); t > 0g in a Banach space has an essential type ! ess such that 1 ! ess ! (where ! is the type of fS(t); t > 0g) and r ess (S(t)) = e !esst ; t > 0 where r ess refers to the essential spectral radius (see e.g. [START_REF] Nagel | One-Parameter Semigroups of Positive Operators[END_REF] p. [START_REF] Schwartz | Analyse IV. Applications de la théorie de la mesure[END_REF][START_REF] Shubin | Spectral theory of the Schrödinger operators on noncompact manifolds: qualitative results[END_REF]. Thus V 2 (T V ) where V is the spectral bound of T V and V is also the type of fU V (t); t > 0g. Note that V 0 by the contraction of the semigroup. A spectral gap for a generator T V refers to

s ess (T V ) < s(T V ) (33)
where s ess (T V ) := sup fRe ; 2 ess (T V )g is the essential spectral bound. We give …rst a weak spectral gap result for generators.

Theorem 33 Let ( ; ; d) be a metric measure space. Let (1 T ) 1 : L 1 ( ) ! L 1 ( ) be weakly compact for any bounded Borel set . We assume that the kernel G 1 (x; y) of (1 T ) 1 satis…es the estimate

sup M >0 lim C!+1 sup y2 Z fx2 M ; d(x;x 0 )>Cg G 1 (x; y) (dx) < 1 1 V (34)
(for some x 0 2 ). Then the peripheral spectrum of T V consists of isolated eigenvalues f k g k with …nite algebraic multiplicities. Moreover, there exists a positive constant " such that for any k, (T V )\D( k ; ") consists of isolated eigenvalues with …nite algebraic multiplicities (D( k ; ") is the disc centered at k with radius ").

It follows that for M and C large enough the norm of c M (1 + iq T V ) 1 + fx2 M ;d(x;x 0 )>Cg (1+iq T V ) 1 is less than 1 1 V : In L 1 spaces, the essential spectrum is stable by weakly compact perturbations (see e.g. [START_REF] Latrach | Essential Spectra on Spaces with the Dunford-Pettis Property[END_REF]) so that

r ess (1 + iq T V ) 1 = r ess h ( c M + fx2 M ;d(x;x 0 )>Cg )(1 + iq T V ) 1 i ( c M + fx2 M ;d(x;x 0 )>Cg )(1 + iq T V ) 1 < 1 1 V = r (1 + iq T V ) 1 :
Since is an isolated eigenvalue of T V with …nite algebraic multiplicity if and only if

1 1+iq
is an isolated eigenvalue of (1 + iq T V ) 1 with …nite algebraic multiplicity, then any spectral value of T V such that

1 j1 + iq j > r ess (1 + iq T V ) 1
is an isolated eigenvalue of T with …nite algebraic multiplicity. Thus, any spectral value of T V such that

j1 + iq j < 1 r ess [(1 + iq T V ) 1 ]
is an isolated eigenvalue of T with …nite algebraic multiplicity. The fact that

j(1 + iq) ( V + iq)j = 1 V < 1 r ess [(1 + iq T V ) 1 ]
shows that V + iq is an isolated eigenvalue of T with …nite algebraic multiplicity; actually there exists a whole disc D( V + iq; ") centered at V + iq (with radius " independent of q) whose intersection with (T V ) consists at most of …nitely many eigenvalues.

The result above shows that there exists an open neighborhood of the axis V + iR whose intersection with (T V ) consists at most of isolated eigenvalues with …nite algebraic multiplicities. However, a priori this neighborhood does not contain a strip of the form f V 6 Re V g since we cannot prevent the existence of a sequence of eigenvalues with real parts going to V and imaginary parts going to in…nity, i.e. [START_REF] Gross | Logarithmic Sobolev Inequalities and contractivity properties of semi-groups[END_REF] does not occur a priori. We derive now a stronger result when fU V (t); t > 0g is norm continuous.

Theorem 34 Let the conditions of Theorem 33 be satis…ed. We assume that fU V (t); t > 0g is norm continuous (e.g. the conditions of Theorem 12 (ii) are satis…ed). Then fU V (t); t > 0g has a spectral gap, i.e. ! ess < V where ! ess is the essential type of fU V (t); t > 0g :

Proof : Let us show …rst that (33) holds. By the norm continuity of fU V (t); t > 0g, we have a Bochner integral

( T V ) 1 = Z +1 0 e t U V (t)dt (Re > V )
(instead of simply a strong integral) so that Riemann-Lebesgue Lemma holds

( T V ) 1 ! 0 as jIm j ! 1:
Suppose now that there exists a sequence of eigenvalues k = k + i k such that k ! V and j k j ! 1 with normalized eigenvectors x k : Then

T V x k = ( k + i k )x k so that (1 + i k T V )x k = (1 k )x k
and then

1 = kx k k = j(1 k )j (1 + i k T V ) 1 x k j(1 k )j (1 + i k T V ) 1
which is impossible if j k j ! 1: This shows (33), i.e. that there exists some > 0 such that (T V ) \ f V 6 Re V g consists of isolated eigenvalues with …nite algebraic multiplicities. Actually the same reasonning shows that this strip cannot contain eigenvalues with imaginary parts as large as we want and …nally this strip contains only …nitely many eigenvalues f 1 ; :::; J g. Let P be the (…nite dimensional) spectral projection corresponding to this …nite set of eigenvalues. Note that this projection commutes with U V (t): We denote by Y its …nite dimensional range. We decompose L 1 ( ) as

L 1 ( ) = X Y
where X = (I P )(L 1 ( )): Then

(T V ) = f 1 ; :::; J g [ (T V jX )
where T V jX is the restriction of T V to X (with domain D(T V ) \ X) and

(T V jX ) = (T V ) \ fRe < V g :
We decompose then U V (t) as

U V (t) = U V (t)P + U V (t)(I P ): It follows that ess (U V (t)) = ess (U V (t)(I P )) (U V (t)(I P ))
where fU V (t)(I P ); t > 0g is identi…ed to the semigroup on X with generator T V jX : Thus

r ess (U V (t)) r (U V (t)(I P )):
Since fU V (t)(I P ); t > 0g is also norm continuous then the spectral mapping theorem

(U V (t)(I P )) f0g = e t (T V jX )
holds (see e.g. [START_REF] Nagel | One-Parameter Semigroups of Positive Operators[END_REF] p 87) so that r (U V (t)(I P )) e ( V )t and …nally we get ! ess < V :

We give now a second (quantitative) approach to spectral gaps for perturbed semigroups relying on the weak type estimate (13):

Theorem 35 Let ( ; ; d) be a metric measure space. We assume that [START_REF] Benci | On a Discreteness Condition of the Spectrum of Schrödinger Operators with Unbounded Potential from Below[END_REF] or ( 14) is satis…ed (e.g. fU (t); t > 0g is holomorphic). Let t > 0 be …xed and let U (t) : L 1 ( ) ! L 1 ( ) be weakly compact for any bounded Borel set . We assume that the kernel p t (x; y) of U (t) satis…es the estimate

sup M >0 lim C!+1 sup y2 Z fx2 M ; d(x;x 0 )>Cg p t (x; y) (dx) < e V t (36)
(for some x 0 2 ). Then ! ess < V ; (more precisely, ! ess where e t is the left hand side of (36)). In particular, for any " < V ! ess , f ; Re > ! ess + "g \ (T V ) is a …nite and non empty set of eigenvalues with …nite algebraic multiplicities.

Proof: Note that (36) is x 0 -independent. We decompose U V (t) as

U V (t) = c M U V (t) + fx2 M ;d(x;x 0 )>Cg U V (t) + fx2 M ;d(x;x 0 )<Cg U V (t) (37)
where c M is the complement of the sublevel set M : Since fx2 M ;d(x;x 0 )<Cg U V (t) is dominated by fx2 M ;d(x;x 0 )<Cg U (t) then, by our assumption, the third operator in (37) is weakly compact. Moreover, by (13) (a consequence of (12)), the norm of c M U V (t) goes to zero as M ! +1: Finally, the norm of fx2 M ;d(x;x 0 )>Cg U V (t) is less than or equal to that of fx2 M ;d(x;x 0 )>Cg U (t) i.e. sup y2 Z fx2 M ; d(x;x 0 )>Cg p t (x; y) (dx):

For M and C large enough, the norm of c M U V (t)+ fx2 M ;d(x;x 0 )>Cg U V (t) is less than e V t : Then the stability of the essential spectrum by weakly compact perturbations in L 1 spaces (see e.g. [START_REF] Latrach | Essential Spectra on Spaces with the Dunford-Pettis Property[END_REF]) shows that

e !esst = r ess [U V (t)] = r ess h ( c M + fx2 M ;d(x;x 0 )>Cg )U V (t) i ( c M + fx2 M ;d(x;x 0 )>Cg )U V (t) < e V t
i.e. ! ess < V . More precisely,

e !esst ( c M + fx2 M ;d(x;x 0 )>Cg )U V (t) 8M; C and letting M ! 1 and C ! 1 yields e !esst sup M >0 lim C!+1 sup y2 Z fx2 M ; d(x;x 0 )>Cg p t (x; y) (dx):
The remaining part follows from standard semigroup theory, see e.g. [START_REF] Nagel | One-Parameter Semigroups of Positive Operators[END_REF].

Actually, the proof of Theorem 35 suggests an interesting alternative.

Corollary 36 Let ( ; ; d) be a metric measure space. We assume that [START_REF] Benci | On a Discreteness Condition of the Spectrum of Schrödinger Operators with Unbounded Potential from Below[END_REF] or ( 14) is satis…ed (e.g. fU (t); t > 0g is holomorphic). Let t > 0 be …xed and let U (t) : L 1 ( ) ! L 1 ( ) be weakly compact for any bounded Borel set . We assume that the kernel p t (x; y) of U (t) satis…es the estimate

sup M >0 lim C!+1 sup y2 Z fx2 M ; d(x;x 0 )>Cg p t (x; y) (dx) < e 1 t
(for some x 0 2 ) where 1 be the spectral bound of T . Then either V < 1 or V = 1 and ! ess < V .

Proof: We have always V 1 : Then either the type of fU V (t); t > 0g (or equivalently the spectral bound of T V ) is strictly less than 1 or then V = 1 and we can of course replace V by 1 in (36) and appeal to Theorem 35.

In particular, if fU (t); t > 0g is a stochastic semigroup (i.e. mass preserving on the positive cone) then R p t (x; y) (dx) = 1 and 1 = 0 so that we have: Corollary 37 Let ( ; ; d) be a metric measure space and let fU (t); t > 0g be a stochastic semigroup (i.e. mass preserving on the positive cone). We assume that [START_REF] Benci | On a Discreteness Condition of the Spectrum of Schrödinger Operators with Unbounded Potential from Below[END_REF] or ( 14) is satis…ed (e.g. fU (t)g is holomorphic). Let t > 0 be …xed and let U (t) : L 1 ( ) ! L 1 ( ) be weakly compact for any bounded Borel set . We assume that the kernel p t (x; y) of U (t) satis…es the estimate

sup M >0 lim C!+1 sup y2 Z fx2 M ; d(x;x 0 )>Cg p t (x; y) (dx) < 1
(for some x 0 2 ). Then either V < 0 or ! ess < V = 0.

Remark 38 If V is bounded then M = for large M: In this case, one shows similarly that if

lim C!+1 sup y2 Z fd(x;x 0 )>Cg p t (x; y) (dx) < e 1 t
then either V < 1 or V = 1 and ! ess < V . Actually, the unperturbed semigroup fU (t); t > 0g itself has a spectral gap and this property is inherited by fU V (t); t > 0g :

Remark 39 We note that if the semigroup fU V (t); t > 0g is irreducible and ! ess < V then V is a strictly dominant (algebraically simple) eigenvalue of T V and lim t!+1 e V t U V (t) P = 0 where P is the spectral projection associated to the leading eigenvalue V (see e.g. [START_REF] Nagel | One-Parameter Semigroups of Positive Operators[END_REF] p. 343-344); in the case V = 0; this implies the so-called "exponential return to equilibrium".

We consider now the case where fU (t); t > 0g operates on all L p ( ) (p > 1); we denote it by fU p (t); t > 0g in L p ( ): We denote by fU pV (t); t > 0g the corresponding perturbed semigroup in L p ( ) and wonder whether the latter has a spectral gap. Let p be the spectral bound of the generator of fU p (t); t > 0g and pV be the spectral bound of the generator of fU pV (t); t > 0g (note that 1V is the above V ). We denote by ! pess the essential type of fU pV (t); t > 0g : Theorem 40 Let ( ; ; d) be a metric measure space and fU (t); t > 0g a contraction semigroup in all L p ( ) (p > 1). We assume that [START_REF] Benci | On a Discreteness Condition of the Spectrum of Schrödinger Operators with Unbounded Potential from Below[END_REF] or ( 14) is satis…ed (e.g. fU (t)g is holomorphic). Let t > 0 be …xed and let U (t) : L 1 ( ) ! L 1 ( ) be compact for any bounded Borel set . We assume that the kernel p t (x; y) of U (t) satis…es the estimate

sup M >0 lim C!+1 sup y2 Z fx2 M ; d(x;x 0 )>Cg p t (x; y) (dx) < e p pt (38)
(for some x 0 2 ). Then either pV < p or pV = p and ! pess < pV :

Remark 41 If the spectral bound p of T p is p-independent (which occurs for some Schrödinger type semigroups [START_REF] Takeda | L p -spectral independence of spectral bounds of Schrödinger type semigroups[END_REF]) then, unless p = 0, Assumption (38) is stronger and stronger as p increases. Does this suggest that the existence of a spectral gap may depend on p ? Note that a priori the spectrum need not be p-independent (see e.g. Laplace Beltrami operator on hyperbolic space [START_REF] Davies | Heat Kernels and Spectral Theory[END_REF] p. 177); this is however the case (in metric spaces with polynomial volume growth and) under Gaussian estimates, see [START_REF] Davies | L p spectral independence and L 1 analyticity[END_REF] and references therein.

We come back to the L 1 -theory. We can avoid the use of the (a priori) unknown parameter V at least for symmetric sub-Markov semigroups. We restrict ourselves to Theorem 35. A basic observation is that the parameter

sup M >0 lim C!+1 sup y2 Z fx2 M ; d(x;x 0 )>Cg p t (x; y) (dx) (39)
depends only on the values of the potential V "at in…nity", (i.e. it is independent of the value of V on the balls B(x 0 ; r) with …nite radius). It su¢ ces then to estimate V by suitable parameters depending on the value of V at "…nite distance" only. In this way, we will be able in principle to check the validity of [START_REF] Hel¤er | Remarks on Decay of Correlations and Witten Laplacians Brascamp-Lieb Inequalities and Semiclassical Limit[END_REF]. We start with:

Lemma 42 Let fU 2 (t); t > 0g be symmetric. Let pV be the spectral bound of T pV the generateur of fU pV (t); t > 0g in L p ( ; ): Then pV V :

Proof: Since the spectral bound V is equal to the type of fU V (t); t > 0g then, for any > V there exists c > 1 such that kU V (t)k L(L 1 ( ; )) c e t and by symmetry kU V (t)k L(L 1 ( ; )) c e t : Then Riesz-Thorin interpolation theorem implies kU pV (t)k L(L p ( ; )) c e t : Since the spectral bound of the generator coincides with the type of the corresponding semigroup for positive semigroups in L p ( ; ) then the type of fU pV (t); t > 0g (or equivalently pV ) is less than or equal to for any > V : Hence pV V :

Thus (36) is satis…ed if sup M >0 lim C!+1 sup y2 Z fx2 M ; d(x;x 0 )>Cg p t (x; y) (dx) < e 2V t
where 2V is the spectral bound of T 2V . It su¢ ces to derive lower bounds of 2V independent of the values of V "at in…nity". Note that T 2V is the self-adjoint operator de…ned by the form

p T 2 ' 2 L 2 ( ; ) + Z V (x) j'(x)j 2 (dx)
with domain

D := ' 2 D( p T 2 ); Z V (x) j'(x)j 2 (dx) < +1 so that 2V = inf '2D; k'k L 2 ( ; ) =1 p T 2 ' 2 L 2 ( ; ) + Z V (x) j'(x)j 2 (dx):
We de…ne D r as the subspace of D consisting of those elements with supports included in B(x 0 ; r): We assume that

9r > 0; D r 6 = f0g (40)
(in our abstract setting, this assumption seems to be necessary). Then D r 6 = f0g for r > r and

2V b r := inf '2Dr; k'k L 2 ( ; ) =1 p T 2 ' 2 L 2 ( ; ) + Z B(x 0 ;r) V (x) j'(x)j 2 (dx):
We note that b r does not depend on the values of V outside the ball B(x 0 ; r) and that r ! b r is nondecreasing. Let b := lim r!+1 b r : (We do not know a priori whether b = 2V .) We note that b and (39) are in some sense independent parameters. Hence we can state:

Theorem

On magnetic Schrödinger operators

As noted in the Introduction, it is possible, in our general formalism, to relax the positivity assumption on the unperturbed semigroup under consideration. Indeed, since a general (i.e. non positivity preserving) contraction semigroup in L 1 spaces admits a modulus contraction semigroup [START_REF] Kipnis | Majoration des semigroupes de contraction de L 1 et applications[END_REF], we could make the appropriate assumptions rather on the modulus of the unperturbed semigroup (or at least on some other dominating semigroup); we do not try to elaborate on this point here in full generality and restrict ourselves to the signi…cant example of Schrödinger operators with magnetic …elds

(r ia) 2 + V = N X j=1 (@ j ia j ) 2 + V (41)
where a = (a 1 ; a 2 ; :::

; a N ) 2 (L 2 loc (R N )) N ; 0 V 2 L 1 loc (R N ):
Following [START_REF] Leinfelder | Schrödinger operators with singular magnetic vector potentials[END_REF], (41) de…nes a self-adjoint operartor H a;V on L 2 (R N ) by means of the closed lower bounded form

h(f; g) = Z R N r a f:r a gdx + Z R N V f gdx
(where r a f := rf iaf is the generalized gradient) with domain

D(h) = n f 2 L 2 (R N ); r a f 2 L 2 (R N ); V 1 2 f 2 L 2 (R N )
o endowed with the norm

kf k h = r kf k 2 + kr a f k 2 + V 1 2 f 2 : It is known ([8] Theorem 2.7) that H a;V is resolvent compact in L 2 (R N ) if H 0;
V is; the key point being the diamagnetic pointwise estimate (42) (see [START_REF] Kondratiev | Discreteness of spectrum and strict positivity criteria for magnetic Schrödinger operators[END_REF] for more recent developements). We provide here an L 1 approach which complements the known hilbertian results. We recall …rst the diamagnetic pointwise estimate [START_REF] Leinfelder | Schrödinger operators with singular magnetic vector potentials[END_REF]:

( H a;V ) 1 f ( H 0;V ) 1 jf j ; f 2 L 2 (R N ) ( > 0): (42)
Exponential formula and (42) imply

e tH a;V f e tH 0;V jf j ; f 2 L 2 (R N ) (43)
we note (symbolically) e tH a;V e tH 0;V : Since e tH 0;V = e t V leaves invariant all L p (R N ) (1 p < +1) then [START_REF] Johnsen | On the spectral properties of Witten-Laplacians, their ranges projections and Brascamp-Lieb's inequality[END_REF] shows that e tH a;V extends as a c 0semigroup

n U a pV (t); t > 0 o in L p (R N ) (1 p < +1) still satisfying the estimate (43) with f 2 L p (R N ):
Theorem 44 We assume that the sublevel sets M are "thin at in…nity" in the sense [START_REF] Davies | Heat Kernels and Spectral Theory[END_REF]:

Then n U a pV (t); t > 0 o is a compact semigroup in L p (R N ) (1 p < +1).
Proof: By Corollary 13 and Theorem 21, e tH 0;V is compact so, [START_REF] Johnsen | On the spectral properties of Witten-Laplacians, their ranges projections and Brascamp-Lieb's inequality[END_REF] in L 1 spaces implies thatfU a 1V (t); t > 0g is weakly compact in L 1 (R N ) by domination. By Dunford-Pettis'theorem (see e.g. [START_REF] Aliprantis | Positive Operators[END_REF] Corollary 5.88, p. 344)

U a 1V (t) = U a 1V ( t 2 )U a 1V ( t 2 ) is compact in L 1 (R N
) as a product of two weakly compact operators. An interpolation argument shows that

n U a pV (t); t > 0 o is compact in all L p (R N ) (1 p < +1).
We show now the existence of a spectral gap when the sublevel sets M are not "thin at in…nity". As far as we know, this problem has not been considered in the literature. This result covers both cases a 6 = 0 (the magnetic case) and a = 0: Theorem 45 Let a V be the spectral bound of H a;V and let a essV be its essential spectral bound in L 2 (R N ). We assume that Proof: The essential spectral bound of H a;V coincides with the essential type of e tH a;V ; t > 0 : We note that

sup M >0 lim C!+1 sup y2R N Z fx2 M ; jxj>Cg 1 (4 t) n 2 exp( jx yj 2 4t )dx < e b t ; (44
a V = inf '2D(h); k'k L 2 =1 Z jr a '(x)j 2 dx + Z V (x) j'(x)j 2 dx b r = inf '2Dr(h); k'k L 2 =1 Z B(0;r) jr a '(x)j 2 dx + Z B(0;r) V (x) j'(x)j 2 dx; and that r(> r) ! b r is nondecreasing so that b a V . In particular sup M >0 lim C!+1 sup y2R N Z fx2 M ; jxj>Cg 1 (4 t) n 2 exp( jx yj 2 4t )dx < e a V t :
We note that the Gaussian estimate behind [START_REF] Johnsen | On the spectral properties of Witten-Laplacians, their ranges projections and Brascamp-Lieb's inequality[END_REF] implies that the (essential) spectrum of H aV is the same in all L p (R N ) (see e.g. [START_REF] Davies | L p spectral independence and L 1 analyticity[END_REF]); in particular, a essV and a V are p-independent. Thus, it su¢ ces to deal with the L 1 case. Let

1 (4 t) n 2 exp( jx yj 2 4t
) be the kernel of e t4 (i.e. the heat kernel on R N ). Estimate [START_REF] Johnsen | On the spectral properties of Witten-Laplacians, their ranges projections and Brascamp-Lieb's inequality[END_REF] gives e tH a;V e tH 0;V e t4

We decompose e tH a;V as e tH a;V = c M e tH a;V + fx2 M ;jxj>Cg e tH a;V + fx2 M ;jxj<Cg e tH a;V

where c M is the complement of the sublevel set M : Since fx2 M ;jxj<Cg e tH a;V fx2 M ;jxj<Cg e t4 and e t4 : L 1 (R N ) ! L 1 loc (R N ) is weakly compact then the third operator in (46) is weakly compact. Moreover, e tH 0;V ; t > 0 is holomorphic in L 1 (R N ) because the heat semigroup is and then e tH 0;V ; t > 0 satis…es the "weak type"estimate (13) so that the operator norm of c M e tH 0;V goes to zero as M ! +1 and this implies that the operator norm of c M e tH a;V goes to zero as M ! +1 as well. Finally, the operator norm of fx2 M ;jxj>Cg e tH a;V is less than or equal to that of fx2 M ;jxj>Cg e t4 i.e.

sup y2R N Z fx2 M ; jxj>Cg 1 (4 t) n 2 exp( jx yj 2 4t
)dx:

Thus, [START_REF] Kato | On the semi-groups generated by Kolmogoro¤'s di¤erential equations[END_REF] shows that for M and C large enough, the operator norm of c M e tH a;V + fx2 M ;jxj>Cg e tH a;V is less than e a V t : On the other hand, since fx2 M ;jxj<Cg e tH a;V is weakly compact then the stability of the essential spectrum by weakly compact perturbations implies e a essV t = r ess e tH a;V = r ess h c M e tH a;V + fx2 M ;jxj>Cg e tH a;V i c M e tH a;V + fx2 M ;jxj>Cg e tH a;V < e a V t and this ends the proof.

On weighted Laplacians

In this section, we revisit some aspects of a topic related to some of the previous results. Let h 2 C 2 (R N ) such that h(x) > 0 8x 2 R N and let in L 2 (R N ; e (x) dx); we do not assume a priori here the …niteness of (dx), i.e. we do not assume that e (x) is integrable. It is well known that

V := 4h h = 1 4 jr (x)j 2 1 2 4 (x):
The (minus) Schrödinger operators

4 := 4 + 1 4 jr j 2 1 2 4
in L 2 (R N ; dx) are also known as the Witten Laplacians (on 0-forms) and were studied in particular in [START_REF] Hel¤er | Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians[END_REF] (the harmonic oscillator) in L 2 (R N ; dx) and is known to generate a compact semigroup. We point out that the Ornstein-Uhlenbeck semigroup is not compact in L 1 (R N ; e jxj 2 2 dx) (see [START_REF] Davies | Heat Kernels and Spectral Theory[END_REF] Section 4.3) while the semigroup generated by (minus) the harmonic oscillator is compact in L 1 (R N ; dx) (see Remark 26(i)).

(ii) We note that by construction 4 + 1 4 jr j 2 1 2 4 is a nonnegative operator in L 2 (R N ; dx) even if 1 4 jr j 2 1 2 4 is not bounded below. Our assumption that 1 4 jr j 2 1 2 4 is bounded below is technical and is due to our L 1 approach of the problem. In ([38] Proposition 3.1, p. 21) we can reach the conclusion of Corollary 47 without this assumption (by means of hypoelliptic techniques) provided there exists some t 2 (1; 2) such that t jr (x)j 2 4 (x) ! +1 as jxj ! 1:

The following (non-convex) potential appears e.g. in [START_REF] Hel¤er | Remarks on Decay of Correlations and Witten Laplacians Brascamp-Lieb Inequalities and Semiclassical Limit[END_REF][43]

(x) = 1 h N X j=1 ( 12
x 4 j + 2 x 2 j ) + 1 h I 2 N X j=1 jx j x j+1 j 2 (47)
(with the convention x N +1 = x 1 ) where h > 0; > 0; < 0; I > 0: so that 4 > 0: On the other hand

jr j 2 = N X j=1 2 4 X j j C (2 j c )x 2 j 1 j i6 =j x 2 i i 3 5 2 > N X j=1 X j j C (2 j c ) 2 x 2(2 j 1) j i6 =j x 4 i i > N X j=1 (2 j c ) 2 x 2(2 j 1) j i6 =j x 4 i i :
We observe that 1 4 jr j 2 1 2 4 > 0 and

n x; 1 4 jr (x)j 2 1 2 4 (x) M o is included in x; x 2(2 j 1) j i6 =j x 4 i i 4M (2 j c ) 2
for any j: It su¢ ces to show that the latter set is thin at in…nity. We may also restrict ourselves to positive coordinates. This set is de…ned by

x j M j i6 =j x 2 i (2 j 1) i where M j = 4M (2 j c ) 2 1 2(2 j 1)

:

To …x the notations, suppose that j = N and set

i := 2 i (2 N 1) ; 1 i N 1:
Note …rst that if a N is large enough then the interesection of a cube

C := fx; a i 1 x i a i + 1; 8ig with the set de…ned by x N M N N 1 i=1 x i i
is empty. On the other hand, it is true that the Lebesgue measure of this interesection is always less than

M N Z a i +1 a 1 1 dx 1 x 1 1
:::

Z a N 1 +1 a N 1 1 dx N 1 x N 1 N 1 = M N 1 (1 1 ) ( 1 (a 1 1) 1 1 (a 1 + 1) 1
) :::

::: 1 (1 N 1 ) ( 1 (a N 1 1) 1 1 (a N 1 + 1) 1 )
Remark 57 One sees that (50) provides us with a su¢ cient condition in terms of sublevel sets of 1 4 jr j 2 1 2 4 for the probability measure (dx) = Z 1 e (x) dx (where Z = R e (x) ) to satisfy the Poincaré inequality.

On Witten Laplacians on 1-forms

The De Rham Complex is given by

d (0) d (1) 0 ! Then (i) T V + + V : D(T V + ) ! L 1 ( ; ) generates a positive semigroup fW (t); t > 0g on L 1 ( ; ). (ii) T V + +V : D(T V + ) ! L 1 ( ; ) is resolvent compact if (16
) is satis…ed (where M are the sublevel sets of V + ).

Proof: (i) The fact T

V + + V : D(T V + ) ! L 1 ( ; ) generates a positive semigroup fW (t); t > 0g on L 1 ( ; ) if and only if lim !+1 r V ( T V + ) 1 < 1
follows from a known result [START_REF] Desch | Perturbations of positive semigroups in AL-spaces[END_REF] (see e.g. [START_REF] Banasiak | Perturbations of Positive Semigroups with Applications[END_REF] Chapter 5).

(ii) For large enough

( T V + V ) 1 = ( T V + ) 1 +1 X i=0 (V ( T V + ) 1 ) i so that T V + + V is resolvent compact if T V + is.
Corollary 66 Let lim !+1 r V ( T V + ) 1 < 1 and let (16) be satis-…ed (where M are the sublevel sets of V + ).

(i) If T generates a holomorphic semigroup then T V + + V generates a (holomorphic) compact semigroup fW (t); t > 0g in L 1 ( ; ).

(ii) We assume that L 1 ( ; ) is separable. Let fU (t); t > 0g be norm continuous and V + be …nite a.e. Let G V + (x; y) be the kernel of (1

T V + ) 1 and let sup y2 Z fV >jg G V + (x; y)V (x) (dx) ! 0 as j ! +1: (53)
Then T V + + V generates a compact semigroup fW (t); t > 0g in L 1 ( ; ).

Proof : (i) We know that T V + generates a holomorphic semigroup [4][41]. The holomorphy of fW (t); t > 0g can be proved for instance as in ( [START_REF] Mokhtar-Kharroubi | On Schrödinger semigroups and related topics[END_REF] Theorem 43). The compactness of fW (t); t > 0g is then deduced as in Theorem 11.

(ii) According to Remark 15 (ii), (1 T ) 1 has a kernel G(x; y): Since (16) implies (15) then (1 T V + ) 1 has also a kernel G V + (x; y): By Theorem 12, T V + generates a norm continuous semigroup. A similar proof shows that T V + + V generates also a norm continuous semigroup and, similarly, the compactness of fW (t); t > 0g is deduced as in Theorem 11.

We observe …rst that (54) implies that T V + + V nc : D(T V + ) ! L 1 ( ) generates a positive semigroup. Secondly [START_REF] Metafune | On the location of the essential spectrum of Schrödinger operators[END_REF]

expresses that V n is T V + -weakly compact or, equivalently, V n is (T V + + V nc )-weakly compact so that lim !+1 r V n ( T V + V nc ) 1 = 0 and T V + + V nc + V n (i.e. T V + + V ) generates a positive semigroup (see [59] Theorem 4). Let s(T V + + V nc ) be the spectral bound of T V + + V nc : We de…ne s := lim n!+1 s(T V + + V nc ):
Note that s(T V + + V nc ) n is nonincreasing and then s > s(T V + ):

We give now an upper bound of the essentiel type of n e t(T V + +V ) ; t > 0 o :

Theorem 69 Let (54)(55) be satis…ed. We assume that either e tT ; t > 0 is holomorphic or e tT ; t > 0 is norm continuous and (53) is satis…ed.

Then the essential type of n e t(T V + +V ) ; t > 0 o is less than or equal to s:

Proof : We know that if e tT ; t > 0 is holomorphic then so are the semigroups n e t(T V + +V nc ) ; t > 0 o and n e t(T V + +V ) ; t > 0 o (see the arguments in the proof of Corollary 67). On the other hand, the inequality Z

fV nc >jg G V + (x; y)V nc (x) (dx) Z fV >jg G V + (x; y)V (x) (dx)
shows that (53) is inherited by V nc so that the two semigroups

n e t(T V + +V nc ) ; t > 0 o and n e t(T V + +V ) ; t > 0 o are norm continuous. In both cases ]0; +1[ 3 t ! R(t) := e t(T V + +V ) e t(T V + +V nc ) is norm continuous. ( 56
)
We note that for large enough

Z 1 0 h e t(T V + +V ) e t(T V + +V nc ) i e t dt = ( T V + V ) 1 ( T V + V nc ) 1 and ( T V + V ) 1 = ( T V + V nc V n ) 1 = ( T V + V nc ) 1 +1 X i=0 (V n ( T V + V nc ) 1 ) i so that Z 1 0 h e t(T V + +V ) e t(T V + +V nc ) i e t dt = ( T V + V nc ) 1 +1 X i=1 (V n ( T V + V nc ) 1 ) i
is weakly compact. By domination, for any t > 0 and " > 0;

Z t+" t h e t(T V + +V ) e t(T V + +V nc ) i e t dt
is weakly compact and …nally, thanks to [START_REF] Mokhtar-Kharroubi | Mathematical Topics in Neutron Transport Theory[END_REF],

h e t(T V + +V ) e t(T V + +V nc ) i e t = lim "!0 " 1 Z t+"
t h e t(T V + +V ) e t(T V + +V nc ) i e t dt is also weakly compact since the limit holds in operator norm. Proof : We note that

V ( T V + ) 1 = V n ( T V + ) 1 + V nc ( T V + ) 1
and (57) expresses that lim n!+1 V nc ( T V + ) 1 L(L 1 ( )) = 0. It follows that V ( T V + ) 1 is weakly compact since V n ( T V + ) 1 is (as a consequence of (55)). Arguing as in the proof of Theorem 70, one sees that R 1 0 h e t(T V + +V ) e tT V + i e t dt is weakly compact and deduce that the difference e t(T V + +V ) e tT V + is also weakly compact for t > 0 which ends the proof of the …rst claim. Finally since the type of n e tT V + ; t > 0 o is less than or equal to that of n e t(T V + +V ) ; t > 0 o then the latter has a spectral gap if the former has. The stability of essential spectral bounds of perturbed generators holds under weaker assumptions: Theorem 72 If (55)(57) are satis…ed then s ess (T V + + V ) = s ess (T V + ): In particular if T V + has a spectral gap then T V + + V has also a spectral gap.

Proof : The …rst claim follows from the weak compactness of the di¤erence ( T V + V ) 1 ( T V + ) 1 and the second claim from the standard inequality s(T V + + V ) > s(T V + ):

We complement Theorem 70 by: Theorem 73 Let the assumptions in Theorem 70 be satis…ed. If

sup n lim !s(T V + +V nc ) r V n ( T V + V nc ) 1 > 1
then the essential type of n e t(T V + +V ) ; t > 0 o is less than its type, i.e. the semigroup n e t(T V + +V ) ; t > 0 o has a spectral gap.

Proof : Suppose there exists n 2 N such that

lim !s(T V + +V nc ) r V n ( T V + V nc ) 1 > 1:
We note that V n (

T V + V nc ) 1 is positive so that r V n ( T V + V nc ) 1 2 (V n ( T V + V nc ));
the weak compactness of V n ( T V + V nc ) 1 implies that r V n ( T V + V nc ) 1 is an eigenvalue of V n ( T V + V nc ) 1 : It follows that s(T V + + V nc ), the spectral bound of T V + + V nc , is (strictly) less than that of

T V + + V nc + V n = T V + + V
and the latter is an eigenvalue of …nite algebraic multiplicity (see [START_REF] Mokhtar-Kharroubi | Mathematical Topics in Neutron Transport Theory[END_REF] Chapter 5 for details). Finally, Theorem 70 implies that the essential type of n e t(T V + +V ) ; t > 0 o is (strictly) less than its type. Similarly, we complement Theorem 71 by: Theorem 74 Let the assumptions in Theorem 71 be satis…ed. If

lim !s(T V + ) r V ( T V + ) 1 > 1 (58)
then the essential type of n e t(T V + +V ) ; t > 0 o is less than its type, i.e. the semigroup n e t(T V + +V ) ; t > 0 o has a spectral gap.

Remark 75 In the context of Theorem 73, (58) implies the existence of a spectral gap for perturbed generators, i.e. s ess (T V + + V ) < s(T V + + V ):

L p theory

For the sake of simplicity (see Remark 82 below) we restrict ourselves to symmetric substochastic semigroups e tT ; t > 0 , i.e. when e tT ; t > 0 coincides with its dual n e tT 0 ; t > 0 o (on L 1 ( )) on the space L 1 ( ) \ L 1 ( ):

Then e tT ; t > 0 interpolates on all L p ( ) (1 p < 1) providing strongly continuous semigroups fU p (t); t > 0g in L p ( ) (where fU 2 (t); t > 0g is selfadjoint in L 2 ( )); we denote their generators by T p (where T 2 self-ajoint). We note that U p;V + (t)f := lim n!+1 e t(Tp V +n ) f (where V +n := min fV + ; ng) de…nes a positive semigroup which is strongly continuous in L p ( ) if and only if U V + (t); t > 0 is strongly continuous in L 1 ( ) (see [START_REF] Voigt | Absorption semigroups, their generators and Schrödinger semigroups[END_REF]); we denote this semigroup by U pV + (t); t > 0 and its generator by T pV + : Moreover U 2V + (t) is self-adjoint with (self-adjoint) generator T 2V + : Under the general assumption

lim !+1 r V ( T V + ) 1 < 1;
one shows (see [START_REF] Mokhtar-Kharroubi | Perturbation theory for convolution semigroups[END_REF]) that the semigroup n e t(T V + +V ) ; t > 0 o on L 1 ( ); with generator

T V + + V : D(T V + ) ! L 1 ( );
interpolates on all L p ( ) (1 p < 1) providing positive strongly continuous semigroups fW p (t); t > 0g = e tAp ; t > 0 in L p ( ) (where A 2 is self-adjoint in L 2 ( )). We point out that V is not a priori pV + -bounded for p > 1: However (see [START_REF] Mokhtar-Kharroubi | New form-bound estimates for many-particle Schrödinger-type Hamiltonians[END_REF]), V is form-bounded with respect to T 2V + with relative form-bound less than or equal to

lim !+1 r V ( T V + ) 1
and A 2 = ( T 2V + ) u ( V ) (form-sum).

Theorem 76 Let lim !+1 r V ( T V + ) 1 < 1 and (16) be satis…ed (where M are the sublevel sets of V + ). Then fW p (t); t > 0g is a compact semigroup in L p ( ) for p > 1:

Proof : We know (see Theorem 66 (ii)) that T V + + V is resolvent compact, i.e. ( T V + V ) 1 is compact in L 1 ( ) for large enough. By interpolation ( A p ) 1 is compact in L p ( ) for large enough. Since fW 2 (t); t > 0g = e tA 2 ; t > 0 is selfadjoint then it is norm continuous so that, by interpolation, e tAp ; t > 0 are norm continuous too for p > 1:

Then arguing e.g. as in the proof of Theorem 17, one sees that e tAp ; t > 0 are compact semigroups when p > 1:

Remark 77 Let R B(x;1) 1
1+V + (y) dy ! 0 as jxj ! +1 and let V 2 L p (R N ) for some p > N 2 : Then the fact that V 2 L p (R N ) with p > N 2 implies that lim !+1 V ( T V + ) 1 L(L 1 (R N )) = 0 (see e.g. [START_REF] Simon | Schrödinger semigroups[END_REF]) shows that (V + V ) generates a compact semigroup in L 1 (R N ). This result should be compared with that in [START_REF] Benci | On a Discreteness Condition of the Spectrum of Schrödinger Operators with Unbounded Potential from Below[END_REF] according to which

+ V + V has a discrete spectrum in L 2 (R N ) if R B(x;1) 1
1+V + (y) dy ! 0 as jxj ! +1 and V 2 L Proof : We know that e t(T V + +V ) e tT V + is weakly compact in L 1 ( ) for t > 0 (see the proof of Theorem 71). In the new notations, e tA 1 e tT 1V + is weakly compact in L 1 ( ) and consequently is square (ii) The existence of a spectral gap of e tAp ; t > 0 when n e tT pV + ; t > 0 o has not a spectral gap is not dealt with here. We note also that it is unclear in general whether a spectral gap of n e t(T V + +V ) ; t > 0 o (see Theorem 75) can be inherited by e tAp ; t > 0 :

h
(ii) We can prove under (55) [START_REF] Mokhtar-Kharroubi | On the strong convex compactness property for the strong operator topology and related topics[END_REF] alone that ( A p ) 1 ( T pV + ) 1 is compact so that A p and T pV + have the same essential spectrum and consequently the same essential spectral bound.

Remark 80 We note that if = R n and if e tT ; t > 0 admits a Gaussian estimate then n e tT V + ; t > 0 o admits also a Gaussian estimate since e tT V + is dominated pointwisely by e tT : On the other hand, the assumption

lim !+1 r V ( T V + ) 1 < 1
remains if we replace V by pV (the product of p and V ) with a suitable p > 1 so that T V + + pV generates a positive semigroup and consequently (see [START_REF] Arendt | Holder's inequality for perturbations of positive semigroups by potentials[END_REF] Theorem 3.6) the semigroup n e t(T V + +V ) ; t > 0 o admits a Gaussian estimate; it follows that the spectrum of e tAp ; t > 0 is p-independent, see e.g. [START_REF] Davies | L p spectral independence and L 1 analyticity[END_REF].

Remark 81 If e tT ; t > 0 is not symmetric but its dual n e tT 0 ; t > 0 o operates also on L 1 ( ) with lim !+1 r h V ( T 0 V + ) 1 i < 1 (L 1 spectral radius) then one can show that n e t(T V + +V ) ; t > 0 o interpolates on all L p ( ) (1 p < 1) providing positive strongly continuous semigroups fW p (t); t > 0g = e tAp ; t > 0 in L p ( ) (in the spirit of [START_REF] Mokhtar-Kharroubi | Perturbation theory for convolution semigroups[END_REF] Theorem 19). If we resume the proof of Theorem 79, we have that R p := e tAp e tT pV + (for p = 1) is weakly compact in L 1 ( ) so that (for j j large enough), ( e tAp ) 1 R p 2 is compact in L p ( ) for all p > 1. Similarly,

h ( e tT pV + ) 1 R p i 2 is compact in L p ( )
for all p > 1. Then the analytic Fredholm alternative shows that e tAp and e tT pV + have the same essential radius, see [START_REF] Voigt | A perturbation theorem for the essential spectral radius of strongly continuous semigroups[END_REF] Corollary 1.4. Finally they share the same essential type.

Theorem 8 1 : 1 o

 811 ) are equivalent. Indeed, a bounded measurable function ' : ( ; ) ! R is said to operate on D(T ) if ' 2 D(T ) fo any 2 D(T ) and 2 D(T ) ! ' 2 D(T ) is continuous for the graph norm. We assume that for each M there exist f M > M and a bounded measurable function ' M : ( ; ) ! R operating on D(T ) with support in f M and equal to one on M : If T V is resolvent compact then for all M , ( T ) 1 : L 1 ( ; ) ! L 1 ( M ; ) is compact. Proof: Let ( T V ) 1 : L 1 ( ; ) ! L 1 ( ; ) be compact. Consider the equation f T f = h; khk L 1 ( ; ) Note …rst that n f ; khk L 1 ( ; ) is bounded in D(T ) endowed with the graph norm. Let ' M : ( ; ) ! R operating on D(T ) with support in f M and equal to one on M : Then

  su¢ ces to show that for any D > 0 sup jyj D Z \fjxj>cg h(x y)dx ! 0 as c ! +1 i.e. sup jyj D Z \fjxj>cg h y (x)dx ! 0 as c ! +1:

  We decompose the integral operator (30) by decomposing its kernel as f (d(x; y)) = 1 c (x)f (d(x; y)) + 1 e c (x)f (d(x; y)) where c := \ fx; d(x; y) > cg and e c := \ fx; d(x; y) < cg since x 2 . Note that f (d(x; y)) f (0) so that ' 2 L 1 ( ; ) ! Z 1 e c (x)f (d(x; y))'(y)u(dy) 2 L 1 ( e c ; ) and (since n e c o is …nite) the imbedding of L 1 ( e c ; ) into L 1 ( e c ; ) is weakly compact because a bounded subset of L 1 ( e c ; ) is equi-integrable. It su¢ ces to show that the norm of the second part goes to zero as c ! +1, i.e. sup y2 Z \fd(x;y)>cg f (d(x; y)) (dx) ! 0 as c ! +1: Consider …rst the integral Z \fd(x;y)>cg

uniformly in y 2 :

 2 The inequality d(y; y) > jd(x; y) d(x; y)j > c (n + 1) for c > (n + 1) shows that either the set fx; n d(x; y) < n + 1g \ fx; d(x; y) > cg is empty (and then [fn d(x; y) < n + 1g \ \ fd(x; y) > cg] = 0) or d(y; y) > c (n + 1): On the other hand, by assumption, for any n [fx; d(x; y) < n + 1g \ ] ! 0 as d(y; y) ! 1 and then (32) follows.As a consequence of Theorem 7, Theorem 10 (ii) and Lemma 31 we have:

)

  where b := lim r!+1 b r and b r = inf '2Dr(h); k'k L 2 =1 Z B(0;r) jr a '(x)j 2 dx + Z B(0;r) V (x) j'(x)j 2 dx where D r (h) are the elements of D(h) with support in the ball B(0; r). Then a essV < a V :

V

  It follows that n e t(T V + +V nc ) ; t > 0 o and n e t(T V + +V ) ; t > 0 o have the same essential spectrum and consequently the same essential type. Note that the essential type of n e t(T V + +V nc ) ; t > 0 o is less than or equal to its type and the latter coincides with the spectral bound s(T V + +V nc ) since n e t(T V + +V nc ) ; t > 0 o is a positive semigroup in L 1 ( ): Finally the essential type of n e t(T V + +V ) ; t > 0 o is less than or equal to s(T V + + V nc ) for all n and this ends the proof. If we replace (54) by lim (x)G V + (x; y) (dx) = 0 (57) then we obtain the following much more precise result: Theorem 70 Let (55)(57) be satis…ed. We assume that either e tT ; t > 0 is holomorphic or e tT ; t > 0 is norm continuous and (53) is satis…ed. Then the essential type of n e t(T V + +V ) ; t > 0 o is equal to that of n e tT V + ; t > 0 o : In particular if n e tT V + ; t > 0 o has a spectral gap then n e t(T V + +V ) ; t > 0 o has also a spectral gap.

Remark 71

 71 We have seen in Section 5 how to estimate the essential type of n e tT V + ; t > 0 o :

N 2 (

 2 R N ): 74 We deal now with spectral stability in L p ( ): Theorem 78 Let (55)(57) be satis…ed. We assume that either e tT ; t > 0 is holomorphic or e tT ; t > 0 is norm continuous and (53) is satis…ed. Then e tAp ; t > 0 and n e tT pV + ; t > 0 o have the same essential spectrum and consequently the same essential type. In particular, e tAp ; t > 0 has a spectral gap if n e tT pV + ; t > 0 o has.

  Theorem 1.6.1, p. 35) we obtain immediately: > 0g is holomorphic) or under the conditions of Theorem 12 (ii), the semigroups fU pV (t); t > 0g are compact in L p ( ; ):If fU 2 (t); t > 0g is self-adjoint, we obtain more precise results: Theorem 17 Let (16) be satis…ed. If fU 2 (t); t > 0g is self-adjoint then the semigroup fU pV (t); t > 0g is compact in L p ( ; ) for p > 1:

	Theorem 16 Let	be -…nite. If (16) or (17) is satis…ed then T pV is
	resolvent compact in L

p ( ; ): Moreover if (12) or (14) are satis…ed (e.g. if fU (t); t

  Let us show now that (19) is satis…ed if is "thin at in…nity". To show (19) it su¢ ces that lim

	still denote by	\fjxj>cg the multiplication operator by the indicator
	function	\fjxj>cg ) because	fjxj>cg f	L 1 ( )
				Z
			jyj!+1	h(x y)dx = 0:

! 0 as c ! +1 uniformly in f in a compact set of L 1 ( ), i.e. (19) holds. Conversely, under (19), H : ' 2 L 1 (R N ) ! L 1 ( ) is a limit in operator norm (as c ! +1) of \fjxj cg H which is compact since \ fjxj cg is bounded.

  [START_REF] Johnsen | On the spectral properties of Witten-Laplacians, their ranges projections and Brascamp-Lieb's inequality[END_REF] Let fU 2 (t); t > 0g be symmetric and (40) be satis…ed. We assume that[START_REF] Benci | On a Discreteness Condition of the Spectrum of Schrödinger Operators with Unbounded Potential from Below[END_REF] or (14) is satis…ed (e.g. fU (t); t > 0g is holomorphic). Let t > 0 be …xed and let U (t) : L 1 ( ) ! L 1 ( ) be weakly compact for any bounded Borel set . If the kernel p t (x; y) of U (t) satis…es the estimate

	sup M >0	lim C!+1	y2 sup	fx2 M ; d(x;x 0 )>Cg Z	p t (x; y) (dx) < e	b t
	then ! ess < V .

  in connection with Fokker-Planck operators. Thus Theorem 46 takes the form: Corollary 47 Let be a real C 2 function on R

	jxj 2 2 dx) unitarily equivalent to (minus) 4+ jxj 2 4	N 2

N : We assume that 1 4 jr j 2 1 2 4 is bounded below. Then the weighted Laplacian 4 on L 2 (R N ; (dx)) (where (dx) = e (x) dx) generates a compact semigroup provided that the sublevel sets of 1 4 jr j 2 1 2 4 are "thin at in…nity".

Remark 48 (i) The Ornstein-Uhlenbeck generator 4 x:r is a weighted Laplacian in L 2 (R N ; e

  e tA 1 e tT 1V + i 2 is compact in L 1 ( ): It follows by interpolation that h e tA 2 e tT 2V + i 2 is compact in L 2 ( ); the self-adjointness of e tA 2 e tT 2V + implies the compactness of e tA 2 e tT 2V + itself in L 2 ( ) and then, by interpolation again, e tAp e tT pV + is compact in L p ( ) for all p > 1: Thus e tAp ; t > 0 and n e tT pV + ; t > 0 o share the same essential spectrum and then the same essential type. The fact that the spectral bound of T pV + is less than or equal to that of A p ends the proof.

Remark 79 (i) We have dealt in Section 5 with spectral gaps of n e tT pV + ; t > 0 o :

Proof : Note that (34) is x 0 -independent. As previously, the existence of the kernel G 1 (x; y) follows from our local weak compactness assumption. Let = V + iq 2 (T V ) be a peripheral spectral value. According to a standard result (see e.g. [START_REF] Nagel | One-Parameter Semigroups of Positive Operators[END_REF] Proposition 2.5, p 67), for any 2 (T V ),

:

In particular the choice of = 1 + iq leads to

We note the standard domination

e t e tT V jf j dt = (1 T V ) 1 jf j :

We decompose (1 + iq T V ) 1 as

where c M is the complement of the sublevel set M : Since fx2 M ;d(x;x 0 )<Cg (1 + iq T V ) 1 is dominated by fx2 M ;d(x;x 0 )<Cg (1 T V ) 1 which is itself dominated by fx2 M ;d(x;x 0 )<Cg (1 T ) 1 then, by our assumption, the third operator in [START_REF] Haslinger | Compactness of the solution operator to @ in weighted L 2 -spaces[END_REF] is weakly compact. Moreover, we saw in the proof of Theorem 7 that the norm of c M (1 T V ) 1 goes to zero as M ! +1 so that, by domination, the norm of c M (1 + iq T V ) 1 goes to zero as M ! +1 as well. Finally, the norm of fx2 M ;d(x;x 0 )>Cg (1 + iq T V ) 1 is less than or equal to that of fx2 M ;d(x;x 0 )>Cg (1 T V ) 1 which is itself less than or equal to that of fx2 M ;d(x;x 0 )>Cg (1 T ) 1 i.e. Proof: We note …rst that pV p : If pV < p there is nothing to prove. Suppose now that pV = p : We resume the idea of proof of Theorem 35 and Corollary 36 but in L p setting with p > 1. We decompose U pV (t) as

where c M is the complement of the sublevel set M : We note the compactness of fx2 M ;d(x;x 0 )<Cg U p (t) in L p ( ) (by interpolation from the L 1 compactness assumption) and then the domination

shows that fx2 M ;d(x;x 0 )<Cg U pV (t) is compact in L p ( ) by Doods-Fremlin's theorem (see e.g. [START_REF] Aliprantis | Positive Operators[END_REF] Theorem 5.20, p. 286). Moreover, by (13) (a consequence of (12)), the L 1 -operator norm of c M U V (t) goes to zero as M ! +1 and its L 1 -operator norm is less than or equal to one. Then, by Riesz-Thorin interpolation theorem, the L p -operator norm of c M U pV (t) goes also to zero as M ! +1. Finally, the L 1 -operator norm of fx2 M ;d(x;x 0 )>Cg U V (t) is less than or equal to that of fx2 M ;d(x;x 0 )>Cg U (t) i.e.

(and its L 1 -operator norm is less than or equal to one) so that, by Riesz-Thorin interpolation theorem, the L p -operator norm of fx2 M ;d(x;x 0 )>Cg U pV (t) is less than or equal to

It follows that for M and C large enough the L p -operator norm of c M U pV (t)+ fx2 M ;d(x;x 0 )>Cg U pV (t) is less than (e p pt ) 1 p = e pt : Then the stability of the essential spectrum by compact perturbations shows that

so that ! pess < pV .

(dx) = h 2 (x)dx: We de…ne the weighted Laplacian

This is (minus) the self-adjoint operator in L 2 (R N ; (dx)) associated to the Dirichlet form R R N jr'j 2 (dx); (see e.g. [START_REF] Davies | Heat Kernels and Spectral Theory[END_REF] Section 4.7, [START_REF] Grigor'yan | Heat kernels on weighted manifolds and applications[END_REF]). It is easy to

Thus the weighted Laplacian 4 in L 2 (R N ; (dx)) is unitarily equivalent to the Schrödinger operator 4 4h h on L 2 (R N ; dx) by the unitary transformation

This shows that the weighted Laplacian 4 in L 2 (R N ; (dx)) has the same spectral properties as the Schrödinger operator 4 4h h on L 2 (R N ; dx); (similar calculations can be performed by replacing the Laplacian by more general elliptic operators with smooth coe¢ cients [START_REF] Davies | L 1 -Properties of Intrinsic Schrödinger semigroups[END_REF] but we restrict ourselves to this model case). We start with the following result already obtained in [START_REF] Metafune | Discreteness of the spectrum for some di¤erential operators with unbounded coe¢ cients in R n[END_REF] by other means.

We assume that 4h h is bounded below. Then the weighted Laplacian 4 generates a compact semigroup on L 2 (R N ; (dx)) provided that the sublevel sets M of 4h h are "thin at in…nity".

Proof: Let V := 4h h . If we consider the operator 4 V then, up to a bounded perturbation, we can assume that V > 0. Then, by Theorem 1, 4 V generates a compact semigroup on L 2 (R N ; dx) and we conclude by a similarity argument.

Generally the function h is written in the form h(x) := e 2 (x) where be a real C 2 function on R N , i.e. (dx) = e (x) dx. Note that in this case

Corollary 49 Let be of the form [START_REF] Krantz | Explorations in Harmonic Analysis with Applications to Complex Function Theory and the Heisenberg Group[END_REF]. Then 4 generates a (holomorphic) compact semigroup in L 1 (R N ; dx):

Proof : Writing (47) in the form

where > 0; > 0; > 0, it is easy to see that

On the other (see [START_REF] Johnsen | On the spectral properties of Witten-Laplacians, their ranges projections and Brascamp-Lieb's inequality[END_REF]) there exists c > 0 such that r (x):x > c jxj 4 for jxj large enough. Thus r (x): x jxj > c jxj 3 and then jr (x)j > c jxj 3 for jxj large enough. Finally

as jxj ! +1 and we are done. Sometimes enjoys useful decompositions. We give a result in this direction and then apply it to uniformly strictly convex : Proof: We note that

We may assume that ( jr 1 j 2 4 1 2 4 1 )+ 1 2 r 1 (x):r 2 (x) and jr 2 (x)j 2 4 1 2 4 2 are nonnegative. One sees that the sublevel sets of jr (x)j 2 4 1 2 4 are included in the sublevel sets jr 2 (x)j 2 4 1 2 4 2 an then are "thin at in…nity" whence 4 generates a (holomorphic) compact semigroup in L 1 (R N ; dx).

A classical result by D. Bakry and M. Emery (see e.g. [START_REF] Royer | Une initiation aux inégalités de Sobolev logarithmiques[END_REF] Théorème 3.1.29, p. 50) asserts that if is uniformly strictly convex with R e (x) dx = 1 then the probability measure (dx) = e (x) dx satis…es a logarithmic-Sobolev (or Gross) inequality and consequently (see e.g. [START_REF] Royer | Une initiation aux inégalités de Sobolev logarithmiques[END_REF] Proposition 3.1.8, p. 37) the spectral gap (or Poincaré) inequality holds. We complement this by the following result which does not rely on the integrability of e (x) : Corollary 51 Let be uniformly strictly convex such that 1 4 jr j 2 1 2 4 is bounded below. Then 4 generates a (holomorphic) compact semigroup in L 1 (R N ; dx): Proof: By assumption, there exists m > 0 such that 00 (x) > mI ( 00 (x) is the Hessian of at x).

I so 1 is uniformly strictly convex and consequently (see e.g. [START_REF] Royer | Une initiation aux inégalités de Sobolev logarithmiques[END_REF]

This ends the proof since jr 2 (x)j 2 4 1 2 4 2 ! +1 as jxj ! 1: We …nd in [START_REF] Hel¤er | Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians[END_REF] systematic results on resolvent compactness or spectral gaps when is a polynomial. In particular, if is a sum of nonpositive monomials then 4 is resolvent compact in L 2 (R N ; dx) if and only if P j j>0 jD x (x)j ! +1 as jxj ! +1; see [START_REF] Hel¤er | Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians[END_REF] Theorem 11.10 (ii), p. 120. We complement this by:

where i > 0 8i for at least one multi-index . Then 4 generates a (holomorphic) compact semigroup in L 1 (R N ).

when i 6 = 1; otherwise replace the corresponding term by ln( (a i +1) (a i +1) ): One sees that

The case of nonnegative polynomials

is much more involved even for homogeneous polynomials, see [START_REF] Hel¤er | Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians[END_REF]. We restrict ourselves to the simplest "elliptic" case.

Proof : It is known (see [START_REF] Hel¤er | Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians[END_REF]) that 1 4 jr j 2 1 2 4 ! +1 as jxj ! 1; this is a consequence of the following facts: The compactness of the unit sphere S N 1 implies the existence of a constant c > 0 such that jr (x)j > c 8x 2 S N 1 and then jr (x)j > c jxj 2r 1 8x 2 R N since is homogeneous of degree 2r; on the other hand, 4 = P j j=r

N . This ends the proof.

Remark 54 (i) Theorem 53 covers e.g. the case

where k > 1:

(ii) When the set of non-zero critical points of (x) is not empty (and under the non-degeneracy condition P 1 j j 2 j@ (x)j 6 = 0 for x 6 = 0) 4 is resolvent compact in L 2 (R N ) provided that for the critical points !, the restriction of 00 (x) to (R!) ? is nondegenerate and not of index 0 (see [START_REF] Hel¤er | Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians[END_REF] Proposition 10.17, p. 108). We conjecture that in this case 4 generates a compact semigroup in [START_REF] Hel¤er | Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians[END_REF] Proposition 10.20, p. 111) that 4 is resolvent compact in L 2 (R 2 ) for all " > 0. We can obtain a stronger conclusion for " > 1. Indeed, on checks that

2" and then, by Remark 26 (i), 4 generates a compact (holomorphic) semigroup in L 1 (R 2 ): Note that 1 4 jr j 2 1 2 4 is not bounded below if " < 1.

We give now an approach of spectral gaps for weighted Laplacians in terms of kernel estimates involving sublevel sets of 1 4 jr j 2 1 2 4 : Two (di¤erent) results on the existence of spectral gaps are given. By adding a positive constant to 1 4 jr j 2 1 2 4 and shifting the Laplacien by if necessary (the heat kernel is then multiplyed by e t ), we may assume without loss of generality that 1 4 jr j 2 1 2 4 is nonnegative.

Theorem 55 Let be a real C 2 function on R N with 1 4 jr j 2 1 2 4 > 0 and let M be the sublevel sets of 1 4 jr j 2 1 2 4 : Then the semigroup generated by the weighted Laplacian on L 2 (R N ; (dx)) has a spectral gap provided that (for some t > 0) sup

where b := lim r!+1 b r and b r is equal to

Proof: By unitary equivalence, we have just to deal with the Schrödinger operator 4 ( 1 4 jr j 2 1 2 4 ) in L 2 (R N ; dx): The corresponding semigroup is dominated by the heat semigroup so that its kernel admits a Gaussian estimate and then (see e.g. [START_REF] Davies | L p spectral independence and L 1 analyticity[END_REF]) its spectrum is the same in all L p spaces. It su¢ ces then to work in the space L 1 (R N ; dx) and use Theorem 43.

In the more usual case where e (x) 2 L 1 (R N ; dx) we obtain a much more precise and explicit result relying on Theorem 35.

Theorem 56 Let be a real C 2 function on R N with 1 4 jr j 2 1 2 4 > 0 and let M be the sublevel sets of 1 4 jr j 2 1 2 4 : We assume that e (x) is integrable. Then the semigroup generated by the weighted Laplacian on L 2 (R N ; (dx)) has a spectral gap provided that (for some t > 0) sup

Proof: When e (x) 2 L 1 (R N ; dx) then (dx) is …nite and then the constant function 1 is an eigenfunction of 4 associated to the eigenvalue 0 which is then the spectral bound of 4 : Then 0 is also the spectral bound of 4 ( 1 4 jr j 2 1 2 4 ) in L 2 (R N ; dx) and also in L 1 (R N ; dx) because the spectrum is the same in L 2 (R N ; dx) and L 1 (R N ; dx) (see e.g. [START_REF] Davies | L p spectral independence and L 1 analyticity[END_REF]) whence V = 0 and we conclude by Theorem 35. we obtain a new Complex, the Witten Complex. By keeping the above L 2 structure we can de…ne closed unbounded operators d (p) : p ! p+1 and Witten Laplacians on p

with 4 (0) = d (0) d (0) : In particular

and 4

(1) = 4

(0)

Id + Hess :

We note that 4 (0) and 4 (1) are lower bounded (nonnegative) operators and C 1 c (the C 1 functions with compact support) is a core for both; see [START_REF] Simader | Essential self-adjointness of Schrödinger operators bounded from below[END_REF].

The Witten Laplacian 4 (0) on 0-forms has been considered in the previous section. The aim of this section is to show that there exist interesting compactness connections between 4 (0) and 4 (1) (see e.g. [START_REF] Johnsen | On the spectral properties of Witten-Laplacians, their ranges projections and Brascamp-Lieb's inequality[END_REF] Theorem 1.3

for other kinds of connections). To this end, we recall …rst a basic functional analytic result related to Glazman's Lemma.

Theorem 58 ([67] Proposition 6.1.4, Corollaries 6.1.1 and 6.1.2, p. 72).

Let A and B be two self-adjoint operators in a Hilbert space H such that

where D H is a core for both A and B: Then: (i) For any real , if (A) \ ( 1; ) is discrete (i.e. consists of isolated eigenvalues of A 1 ) then (B) \ ( 1; ) is also discrete (i.e. consists of isolated eigenvalues of B).

(ii) If we denote by

their eigenvalues in ( 1; ); numbered according to their multiplicities, then A k B k :

For any bounded below self-adjoint operator A, we de…ne its essential lower bound as ess = sup f 2 R; (A) \ ( 1; ) is discreteg : We start with two theorems based on a convexity assumption.

Theorem 59 We assume that is a convex C 2 function. Let 0 ess and 1 ess be respectively the essential lower bound of 4 (0) and 4 (1) : Then 0 ess 1 ess : In particular, if 4 (0) is resolvent compact then 4 (1) is also resolvent compact.

Proof:

Id and B = 4 (1) : The convexity of implies that Hess is a form-nonnegative multiplication (matrix) operator so that (A!; !) (B!; !) for C 1 c 1-forms !: Note that A is nothing but N copies of 4

(0) so that A has the same spectral strucure as 4 (0) : In particular, the essential lower bound of 4 (0) coincides with that of A: Thus (A) \

( 1; 0 ess ) is discrete and then, by Theorem 58, (B) \ ( 1; 0 ess ) is also discrete so that 0 ess 1 ess : In particular, if 4 (0) is compact then 0 ess = +1 and then (B) is purely discrete, i.e. B is resolvent compact; (this last property holds without convexity assumption, see Theorem 63 below).

Remark 60 Note that below the value 0 ess , the k-th eigenvalue of 4 (0) is majorized by that of 4 (1) :

We provide now a strict inequality between the spectral bottoms of 4

and 4 (1) :

Theorem 61 Let be a convex C 2 function and let 0 and 1 be respectively the spectral bottom of 4

(0) and 4 (1) : We assume that 0 is an isolated eigenvalue of 4 (0) : If the lowest eigenvalue (x) of Hess (x) is not identically zero then 1 > 0 :

Proof : We note that for a 1-form !(x) = P N j=1 ! j (x)dx j we have

i.e. Hess > (x)Id: It follows that

and then the spectral bottom of (4 (0) + ) Id (or equivalently the spectral bottom of 4 (0) + ) is less than or equal to the spectral bottom of 4 (1) :

It su¢ ces then to compare 0 the the spectral bottom e 0 of 4 (0) + : By the convexity of we have > 0 and then 4

+ implying the trivial inequality 0 e 0 : Suppose now that 0 is an isolated eigenvalue of 4 (0) ; then there exists > 0 such that (4 (0) ) \ 0 ; 0 + is discrete and then, by Theorem 58, (4 (0) + ) \ 0 ; 0 + is also discrete (possibly empty). Thus, if e 0 > 0 + we are done. Otherwise, e 0 is an isolated eigenvalue; by a classical result it is simple and is associated to a normalized positive (almost everywhere) eigenfunction e f : By assumption, there exists also a normalized positive (almost everywhere) eigenfunction f associated to the eigenvalue 0 of 4 (0) : The fact that (f; e f ) > 0 when (:) is not identically zero implies

Remark 62 In the case where R e (x) dx = 1 then 0 = 0 and, by Theorem 61, 1 > 0 so that 4 (1) is invertible allowing thus the derivation of the "exact" Hel¤ er-Sjöstrand's covariance formula while Brascamp-Lieb's inequality Z (f (x) hf i)(g(x) hgi)e (x) dx (Hess ) 1 df; dg is meaningful for strictly convex only; see [START_REF] Johnsen | On the spectral properties of Witten-Laplacians, their ranges projections and Brascamp-Lieb's inequality[END_REF] for more information.

We give a compactness result for 4 (1) which does not rely on a convexity assumption.

Theorem 63 We assume that is a C 2 function. Let (x) be the lowest eigenvalue of Hess (x): We assume that 1 4 jr j 2 1 2 4 + (x) is bounded below. Then 4 (1) is resolvent compact if 4 + 1 4 jr j 2 1 2 4 + (x) is. In particular, if both 1 4 jr j 2 1 2 4 and (x) are bounded below then 4

is resolvent compact provided that 4 (0) is resolvent compact or the sublevel sets of (x) are thin at in…nity.

Proof : It follows from [START_REF] Lenz | Compactness of Schrödinger semigroups[END_REF] and Theorem 58, that 4 (1) is resolvent compact if 4 (0) + (x) is; the remainder is clear.

We deal now with spectral gaps for Witten Laplacians on 1-forms. We assume that is a C 2 function such that 1 4 jr j 2 1 2 4 + is bounded below; for simplicity, we assume that 1 4 jr j 2 1 2 4 + > 0 (otherwise we "shift"the operator by adding a suitable constant). Let D 1 be the space of 1-form

We note that the bottom of the spectrum of 4 (1) is

while the bottom of the spectrum of 4

where D 1 r are the elements of D 1 with support in B(0; r): We note that We denote by M the sublevel sets of 1 4 jr j 2 1 2 4 + : We assume that for some t > 0 we have

Then 4

(1) has a spectral gap.

Proof: Let 0 ess be the essential lower bound of 4 (0) + (x): By resuming the proofs of Theorem 35, one sees that (52) implies that the essential type of the semigroup generated by (4

and Theorem 58 show that (4 (1) ) \ ( 1; 0 ess ) is discrete. The fact that 0 ess > 1 shows that 4

(1) has a spectral gap.

On inde…nite potentials

This last section continues the general theory of Section 2 and deals with inde…nite potentials V = V + V , i.e. we consider the more general case

where T is the generator of a substochastic semigroup fU (t); t > 0g in L 1 ( ; ) and V are nonnegative and unbounded and are not necessarily the positive and negative parts of V .

L 1 theory

We give …rst a meaning to "T V + + V ". Our general assumption is that V + is admissible for fU (t); t > 0g, i.e. U V + (t); t > 0 is a c 0 -semigroup; its generator is then denoted by T V + (see the beginning of Section 2).

Theorem 65 Let V : D(T V + ) ! L 1 ( ; ) be T V + -bounded and let

Remark 67 (i) See Theorem 28 or Theorem 32 (i) to check [START_REF] Davies | Quantum dynamical semigroups and the neutron di¤usion equation[END_REF] in metric measure spaces.

(ii) By the domination (1 T V + ) 1 (1 T ) 1 we may replace G V + (x; y) by G(x; y) in (53):

Remark 68 The semigroup generated by T V + + V (under the general assumption lim !+1 r V ( T V + ) 1 < 1) is given by a Feynmann-Kac formula and is attached intrinsically to V (i.e. it is independent of the choice of a decomposition V = V + V ), see [START_REF] Mokhtar-Kharroubi | Perturbation theory for convolution semigroups[END_REF] Remark 16.

The above compactness results are due to the part V + of the potential. In particular, it is not possible that ( T V + V ) 1 be compact if ( T V + ) 1 is not ! This stems from the fact that the domination

would imply the weak compactness of ( T V + ) 1 and in fact its compactness (see Lemma 6). Similarly n e t(T V + +V nc ) ; t > 0 o cannot be compact if n e tT V + ; t > 0 o is not. On the other hand, regardless of V + , the part V may induce di¤erent compactness results with di¤erent spectral consequences. To this end, we assume now that is a locally compact metric space endowed with a locally …nite Borel measure > 0: Let n be a nondecreasing sequence of compact subsets such that = [ n n and let

V n := V on n 0 on c n :

We note that V = V n + V nc . Our …rst assumption is

this expresses simply that lim n!+1 V nc ( T V + ) 1 L(L 1 ( )) < 1: Our second assumption is that for any compact sets C