Valentin A Zagrebnov
email: valentin.zagrebnov@univ-amu.fr

Trotter-Kato Product Formulae in Dixmier Ideal

It is shown that for a certain class of the Kato functions, the Trotter-Kato product formulae converge in Dixmier ideal C 1,∞ in topology, which is defined by the • 1,∞ -norm. Moreover, the rate of convergence in this topology inherits the error-bound estimate for the corresponding operator-norm convergence.

Preliminaries. Symmetrically-Normed Ideals

.

.

. The authors considered the case of the Gibbs-Schrödinger semigroups. They scrutinised, in these papers, a dependence of the rate of conver-

gence for the (exponential) Trotter formula on the smoothness of the potential in the Schrödinger generator.

The first abstract result in this direction was due to [START_REF] Neidhardt | Trotter-Kato product formula and symmetrically normed ideals[END_REF]. In this paper, a general scheme of lifting the operator-norm rate convergence for the Trotter-Kato product formulae was proposed and advocated for estimating the rate of the trace-norm convergence. This scheme was then improved and extended in [START_REF] Cachia | Trotter product formula for nonself-adjoint Gibbs semigroups[END_REF] to the case of nonself-adjoint Gibbs semigroups.

The aim of the present note is to elucidate the question about the existence of other than the von Neumann-Schatten proper two-sided ideals I(H ) of L (H ) and then to prove the (non-exponential) Trotter-Kato product formula in topology of these ideals together with an estimate of the corresponding rate of convergence. Here, a particular case of the Dixmier ideal C 1,∞ (H ) [START_REF] Connes | Noncommutative Geometry[END_REF][START_REF] Dixmier | Von Neumann Algebras[END_REF] is considered. To specify this ideal we recall in Section 2, the notion of singular trace and then of the Dixmier trace [START_REF] Carey | Dixmier traces and some applications in non-commutative geometry[END_REF][START_REF] Dixmier | Existence des traces non normales[END_REF] in Section 3. Main results about the Trotter-Kato product formulae in the Dixmier ideal C 1,∞ (H ) are collected in Section 4. There the arguments based on the lifting scheme [START_REF] Neidhardt | Trotter-Kato product formula and symmetrically normed ideals[END_REF] (Theorem 5.1) are refined for proving the Trotter-Kato product formulae convergence in the • 1,∞ -topology with the rate, which is inherited from the operator-norm convergence.

To this end, in the rest of the present section, we recall an important auxiliary tool: the concept of symmetrically-normed ideals, see e.g. [START_REF] Gohberg | Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space[END_REF][START_REF] Simon | Trace Ideals and Their Applications[END_REF].

Let c 0 ⊂ l ∞ (N) be the subspace of bounded sequences ξ = {ξ j } ∞ j=1 ∈ l ∞ (N) of real numbers, which tend to zero. We denote by c f the subspace of c 0 consisting of all sequences with finite number of non-zero terms (finite sequences). Definition 1 A real-valued function φ : ξ → φ(ξ ) defined on c f is called a norming function if it has the following properties: φ(ξ ) > 0, ∀ξ ∈ c f , ξ = 0, (1.1)

φ(αξ ) = |α|φ(ξ ), ∀ξ ∈ c f , ∀α ∈ R, (1.2)
φ(ξ + η) ≤ φ(ξ ) + φ(η), ∀ξ, η ∈ c f , (1.3)
φ(1, 0, . . .) = 1.

(1.4)

A norming function φ is called to be symmetric if it has the additional property φ(ξ 1 , ξ 2 , ..., ξ n , 0, 0, . . .) = φ(|ξ j 1 |, |ξ j 2 |, ..., |ξ j n |, 0, 0, . . .) (1.5) for any ξ ∈ c f and any permutation j 1 , j 2 , . . . , j n of integers 1, 2, . . . , n.

It turns out that for any symmetric norming function and for any elements ξ, η from the positive cone c + of non-negative, non-increasing sequences such that ξ, η ∈ c f obey ξ 1 ≥ ξ 2 ≥ . . . ≥ 0, η 1 ≥ η 2 ≥ . . . ≥ 0 and n j=1 ξ j ≤ n j=1 η j , n = 1, 2, . . . , (1.6) valentin.zagrebnov@univ-amu.fr one gets the Ky Fan inequality [START_REF] Gohberg | Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space[END_REF] (Sect. 3, §3) :

φ(ξ ) ≤ φ(η).

(1.7)

Moreover, (1.7) together with the properties, (1.1), (1.2) and (1.4) imply Therefore, any symmetric norming function φ is uniquely defined by its values on the positive cone c + . Now, let ξ = {ξ 1 , ξ 2 , . . .} ∈ c 0 . We define

ξ 1 ≤ φ(ξ ) ≤ ∞ j=1 ξ j , ξ ∈ c + f := c f ∩ c + . ( 1
ξ (n) := {ξ 1 , ξ 2 , . . . , ξ n , 0, 0, . . . } ∈ c f .
Then, if φ is a symmetric norming function, we set

c φ := {ξ ∈ c 0 : sup n φ(ξ (n) ) < +∞}. (1.12)
Therefore, one gets

c f ⊆ c φ ⊆ c 0 .
Note that by (1.5)-(1.7) and (1.12), one gets

φ(ξ (n) ) ≤ φ(ξ (n+1) ) ≤ sup n φ(ξ (n) )
, for any ξ ∈ c φ .

Then, the limit φ(ξ )

:= lim n→∞ φ(ξ (n) ) , ξ ∈ c φ ,
exists and φ(ξ ) = sup n φ(ξ (n) ), i.e. the symmetric norming function φ is a normal functional on the set c φ (1.12), which is a linear space over R. By (1.3) and (1.10) one also gets that any symmetric norming function is continuous on c f :

|φ(ξ ) -φ(η)| ≤ φ(ξ -η) ≤ φ 1 (ξ -η) , ∀ξ, η ∈ c f .
Suppose that X is a compact operator, i.e. X ∈ C ∞ (H ). Then by s(X ) := {s 1 (X ), s 2 (X ), . . . }, we denote the sequence of singular values of X counting multiplicities. We always assume that

s 1 (X ) ≥ s 2 (X ) ≥ . . . ≥ s n (X ) ≥ . . . .
To define symmetrically-normed ideals of the compact operators C ∞ (H ), we introduce the notion of a symmetric norm.

Definition 2

Let I be a two-sided ideal of C ∞ (H ). A functional • sym : I → R + 0 is called a symmetric norm if besides the usual properties of the operator norm • :

X sym > 0, ∀X ∈ I, X = 0, α X sym = |α| X sym , ∀X ∈ I, ∀α ∈ C, X + Y sym ≤ X sym + Y sym , ∀X, Y ∈ I,
it verifies the following additional properties:

AX B sym ≤ A X sym B , X ∈ I, A, B ∈ L (H ), (1.13) α X sym = |α| X = |α| s 1 (X ), for any one -rank operator X ∈ I. (1.14)
If the condition (1.13) is replaced by

U X sym = XU sym = X sym , X ∈ I , (1.15)
for any unitary operator U on H , then, instead of the symmetric norm, one gets definition of an invariant norm • inv .

First, we note that the ordinary operator norm • on any ideal I ⊆ C ∞ (H ) is evidently a symmetric norm as well as an invariant norm.

Second, we observe that, in fact, every symmetric norm is invariant. Indeed, for any unitary operators U and V one gets by (1.13) that U X V sym ≤ X sym , X ∈ I .

(1.16)

Since X = U -1 U X V V -1
, we also get X sym ≤ U X V sym , which together with (1.16) yield (1.15). Third, we claim that X sym = X * sym . To this aim, let X = U |X | be the polar representation of the operator X ∈ I. Since U * X = |X |, then by (1.13), we obtain X sym = |X | sym . The same line of reasoning applied to the adjoint operator X * = |X |U * yields X * sym = |X | sym , which proves the claim. Now, we can apply the concept of the symmetric norming functions to describe the symmetrically-normed ideals of the unital algebra of bounded operators L (H ), or in general, the symmetrically-normed ideals generated by symmetric norming functions. Recall that any proper two-sided ideal I(H ) of L (H ) is contained in compact operators C ∞ (H ) and contains the set K (H ) of finite-rank operators, see e.g. [START_REF] Pietsch | Traces of operators and their history[END_REF][START_REF] Simon | Trace Ideals and Their Applications[END_REF]:

K (H ) ⊆ I(H ) ⊆ C ∞ (H ).
(1.17)

To clarify the relation between symmetric norming functions and the symmetricallynormed ideals, we mention that there is an obvious one-to-one correspondence between functions φ (Definition 1) on the cone c + and the symmetric norms • sym on K (H ). To proceed with a general setting, one needs definition of the following relation.

Definition 3

Let c φ be the set of vectors (1.12) generated by a symmetric norming function φ. We associate with c φ a subset of compact operators .18) This definition implies that the set C φ (H ) is a proper two-sided ideal of the algebra L (H ) of all bounded operators on H . Setting

C φ (H ) := {X ∈ C ∞ (H ) : s(X ) ∈ c φ } . ( 1
X φ := φ(s(X )) , X ∈ C φ (H ) , (1.19)
one obtains a symmetric norm:

• sym = • φ , on the ideal I = C φ (H ) (Definition 2
) such that this symmetrically-normed ideal becomes a Banach space. Then, in accordance with (1.17) and (1.18), we obtain by (1.10) that

K (H ) ⊆ C 1 (H ) ⊆ C φ (H ) ⊆ C ∞ (H ) . (1.20)
Here, the trace-class operators C 1 (H ) := C φ 1 (H ), where the symmetric norming function φ 1 is defined in (1.9), and

X φ ≤ X 1 , X ∈ C 1 (H ) .
φ (r ) (ξ ) := r j=1 ξ * j , ξ ∈ c f ,
generates for arbitrary fixed r ∈ N, the symmetrically-normed ideals, which are trivial in the sense that all C φ (r ) (H ) = C ∞ (H ). Criterion for an operator A to belong to a nontrivial ideal C φ is

M = sup m≥1 P m A P m φ < ∞ , (1.21)
where {P m } m≥1 is a monotonously increasing sequence of the finite-dimensional orthogonal projectors on H strongly convergent to the identity operator [START_REF] Gohberg | Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space[END_REF]. Note that for A ∈ C ∞ , the condition (1.21) is trivial.

We consider now a couple of examples to elucidate the concept of the symmetricallynormed ideals C φ (H ) generated by the symmetric norming functions φ and the rôle of the functional trace on these ideals.

Example 1 The von Neumann-Schatten ideals C p (H ) [START_REF] Schatten | Norm Ideals of Completely Continuous Operators[END_REF]. These ideals of C ∞ (H ) are generated by symmetric norming functions φ(ξ ) := ξ p , where

ξ p =   ∞ j=1 |ξ j | p   1/ p , ξ ∈ c f ,
for 1 ≤ p < +∞, and by

ξ ∞ = sup j |ξ j |, ξ ∈ c f ,
for p = +∞. Indeed, if we set {ξ * j := s j (X )} j≥1 , for X ∈ C ∞ (H ), then the symmetric norm X φ = s(X ) p coincides with X p and the corresponding symmetricallynormed ideal C φ (H ) is identical to the von Neumann-Schatten class C p (H ).

By definition, for any X ∈ C p (H ) the trace: |X | → Tr|X | = j≥1 s j (X ) ≥ 0. The trace norm X 1 = Tr|X | is finite on the trace-class operators C 1 (H ) and it is valentin.zagrebnov@univ-amu.fr infinite for X ∈ C p>1 (H ). We say that for p > 1, the von Neumann-Schatten ideals admit no trace, whereas for p = 1 the map: X → Tr X exists and it is continuous in the • 1 -topology.

Note that by virtute of the Tr-linearity the trace norm:

C 1,+ (H ) ∋ X → X 1 is linear on the positive cone C 1,+ (H ) of the trace-class operators.
Example 2 Now, we consider symmetrically-normed ideals C Π (H ). To this aim let Π = {π j } ∞ j=1 ∈ c + be a non-increasing sequence of positive numbers with π 1 = 1. We associate with Π the function

φ Π (ξ ) = sup n    1 n j=1 π j n j=1 ξ * j    , ξ ∈ c f . (1.22)
It turns out that φ Π is a symmetric norming function. Then, the corresponding (1.12) set c φ Π is defined by

c φ Π :=    ξ ∈ c f : sup n 1 n j=1 π j n j=1 ξ * j < +∞    .
Hence, the two-sided symmetrically-normed ideal C Π (H ) := C φ Π (H ) generated by symmetric norming function (1.22) consists of all those compact operators X that

X φ Π := sup n 1 n j=1 π j n j=1 s j (X ) < +∞ . (1.23)
This equation defines a symmetric norm

X sym = X φ Π on the ideal C Π (H ), see Definition 2. Now, let Π = {π j } ∞ j=1 , with π 1 = 1, satisfy ∞ j=1 π j = +∞ and lim j→∞ π j = 0 . (1.24) Then, the ideal C Π (H ) is nontrivial: C Π (H ) = C ∞ (H ) and C Π (H ) = C 1 (H ),
see Remark 2, and one has

C 1 (H ) ⊂ C Π (H ) ⊂ C ∞ (H ). (1.25)
If in addition to (1.24), the sequence Π = {π j } ∞ j=1 is regular, i.e. it obeys

n j=1 π j = O(nπ n ) , n → ∞ , (1.26) then X ∈ C Π (H ), if and only if s n (X ) = O(π n ) , n → ∞ , (1.27)
cf. condition (1.21). On the other hand, the asymptotics

s n (X ) = o(π n ) , n → ∞ ,
implies that X belongs to some ideal in between of C 1 (H ) and C 1+ε (H ).

Remark 3 A natural choice of the sequence {π j } ∞ j=1 that satisfies (1.24) is π j = j -α , 0 < α ≤ 1. Note that if 0 < α < 1, then the sequence Π = {π j } ∞
j=1 satisfies (1.26), i.e. it is regular for ε = 1 -α. Therefore, the two-sided symmetrically-normed ideal C Π (H ) generated by symmetric norming function (1.22) consists of all those compact operators X , which singular values obey (1.27):

s n (X ) = O(n -α ), 0 < α < 1, n → ∞ . (1.28) Let α = 1/ p , p > 1.
Then, the corresponding to (1.28) symmetrically-normed ideal is defined by

C p,∞ (H ) := {X ∈ C ∞ (H ) : s n (X ) = O(n -1/ p ), p > 1} ,
which is known as the weak-C p ideal [START_REF] Pietsch | Traces of operators and their history[END_REF][START_REF] Simon | Trace Ideals and Their Applications[END_REF].

Whilst by virtue of (1.28) the weak-C p ideal admit no trace, definition (1.23) implies that for the regular case p > 1 a symmetric norm on C p,∞ (H ) is equivalent to

X p,∞ = sup n 1 n 1-1/ p n j=1 s j (X ) , (1.29)
and

C 1 (H ) ⊂ C p,∞ (H ) ⊂ C ∞ (H ).

Singular Traces

Note that (1.29) implies:

C 1 (H ) ∋ A → A p,∞ < ∞,
but any related linear, positive and unitarily invariant functional (trace), which is nontrivial on the ideal C p,∞ (H ) is zero on the set of finite-rank operators K (H ). We remind that these non-normal traces:

Tr ω (X ) := ω({n -1+1/ p n j=1 s j (X )} ∞ n=1 ) , (2.1)
valentin.zagrebnov@univ-amu.fr are called singular, [START_REF] Dixmier | Existence des traces non normales[END_REF][START_REF] Lord | Singular Traces. Theory and Applications[END_REF]. Here, ω is an appropriate linear positive normalised functional (state) on the Banach space l ∞ (N) of bounded sequences. Recall that the set of these states S (l ∞ (N)) ⊂ (l ∞ (N)) * , where (l ∞ (N)) * is dual of the Banach space l ∞ (N). Note that the singular trace (2.1) is continuous in topology defined by the norm (1.29).

Remark 4 (a) The weak-C p ideal, which is defined for p = 1 by

C 1,∞ (H ) := {X ∈ C ∞ (H ) : n j=1 s j (X ) = O(ln(n)), n → ∞} , (2.2)
has a special interest. Note that since Π = { j -1 } ∞ j=1 does not satisfy (1.26), the characterisation s n (X ) = O(n -1 ) is not true, see (1.27), (1.28). In this case, the equivalent norm can be defined on the ideal (2.2) as

X 1,∞ := sup n∈N 1 1 + ln(n) n j=1 s j (X ).
(2.3) By virtute of (1.25) and Remark 3 one gets that

C 1 (H ) ⊂ C 1,∞ (H ) ⊂ C 1+ε (H ) for any ε > 0 and that C 1 (H ) ∋ A → A 1,∞ = A 1 ≥ A 1+ε . (b) In contrast to the trace norm • 1 on the positive cone C 1,+ (H ), see Example 1, the map X → X 1,∞ on the positive cone C 1,∞,+ (H )
is not linear although homogeneous. On the space l ∞ (N), there exists a state ω ∈ S (l ∞ (N)) such that the map

X → Tr ω (X ) := ω({(1 + ln(n)) -1 n j=1 s j (X )} ∞ n=1 ) , (2.4)
is linear and verifies the properties of the (singular) trace for any X ∈ C 1,∞ (H ).

We construct ω in Section 3. This particular choice of the state ω defines the Dixmier trace on the space C 1,∞ (H ), which is called, in turn, the Dixmier ideal, see e.g. [START_REF] Carey | Dixmier traces and some applications in non-commutative geometry[END_REF][START_REF] Connes | Noncommutative Geometry[END_REF]. The Dixmier trace (2.4) is obviously continuous in topology defined by the norm (2.3). This last property is basic for discussion in Section 4 of the Trotter-Kato product formula in the • p,∞ -topology, for p ≥ 1.

Example 3 With non-increasing sequence of positive numbers π = {π j } ∞ j=1 , π 1 = 1, one can associate the symmetric norming function φ π given by

φ π (ξ ) := ∞ j=1 π j ξ * j , ξ ∈ c f .
The corresponding symmetrically-normed ideal is denoted by C π (H ) := C φ π (H ).

If the sequence π satisfies (1.24), then ideal C π (H ) does not coincide neither with C ∞ (H ) nor with C 1 (H ). If, in particular, π j = j -α , j = 1, 2, . . . , for 0 < α ≤ 1, then the corresponding ideal is denoted by C ∞, p (H ), p = 1/α. The norm on this ideal is given by

X ∞, p := ∞ j=1 j -1/ p s j (X ) , p ∈ [1, ∞).
The symmetrically-normed ideal C ∞,1 (H ) is called the Macaev ideal [START_REF] Gohberg | Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space[END_REF]. It turns out that the Dixmier ideal

C 1,∞ (H ) is dual of the Macaev ideal: C 1,∞ (H ) = C ∞,1 (H ) * . Proposition 2.1 The space C 1,∞ (H ) endowed with the norm • 1,∞ is a Banach space.
The proof is quite standard although tedious and long. We address the readers to the corresponding references, e.g. [START_REF] Gohberg | Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space[END_REF].

Proposition 2.2 The space C 1,∞ (H ) endowed with the norm • 1,∞ is a Banach ideal in the algebra of bounded operators L (H ).
Proof To this end, it is sufficient to prove that if A and C are bounded operators, then

B ∈ C 1,∞ (H ) implies ABC ∈ C 1,∞ (H ).
Recall that singular values of the operator ABC verify the estimate s j (ABC) ≤ A C s j (B). By (2.3), it yields

ABC 1,∞ = sup n∈N 1 1 + ln(n) n j=1 s j (ABC) ≤ (2.5) A C sup n∈N 1 1 + ln(n) n j=1 s j (B) = A C B 1,∞ ,
and consequently the proof of the assertion.

Recall that for any A ∈ L (H ) and all B ∈ C 1 (H ) one can define a linear functional on C 1 (H ) given by Tr H (AB). The set of these functionals {Tr

H (A•)} A∈L (H ) is just the dual space C 1 (H ) * of C 1 (H ) with the operator-norm topology. In other words, L (H ) = C 1 (H ) * , in the sense that the map A → Tr H (A•) is the isometric isomorphism of L (H ) onto C 1 (H ) * .
With the help of the duality relation

A|B := Tr H (AB) , (2.6
ideal: C 1,∞ (H ) = C ∞,1 (H ) * , where C ∞,1 (H ) = {A ∈ C ∞ (H ) : n≥1 1 n s n (A) < ∞} , (2.7)
see Example 3. By the same duality relation and by similar calculations one also obtains that the

predual of C ∞,1 (H ) is the ideal C ∞,1 (H ) * = C (0)
1,∞ (H ), defined by

C (0) 1,∞ (H ) := {A ∈ C ∞ (H ) : n j≥1 s j (A) = o(ln(n)), n → ∞} . (2.8)
By virtue of (2.2) (see Remark 4), the ideal (2.8) is not self-dual since

C (0) 1,∞ (H ) * * = C 1,∞ (H ) ⊃ C (0) 1,∞ (H ).
The problem which has motivated construction of the Dixmier trace in [START_REF] Dixmier | Existence des traces non normales[END_REF] was related to the question of a general definition of the trace, i.e. a linear, positive and unitarily invariant functional on a proper Banach ideal I(H ) of the unital algebra of bounded operators L (H ). Since any proper two-sided ideal I(H ) of L (H ) is contained in compact operators C ∞ (H ) and contains the set K (H ) of finite-rank operators ((1.17), Section 1), domain of definition of the trace has to coincide with the ideal I(H ). Note that every nontrivial normal trace on L (H ) is proportional to the canonical trace Tr H (•), see e.g. [START_REF] Dixmier | Von Neumann Algebras[END_REF][START_REF] Pietsch | Traces of operators and their history[END_REF]. Therefore, the Dixmier trace (2.4

) : C 1,∞ ∋ X → Tr ω (X ), is not normal. Definition 4 A trace on the proper Banach ideal I(H ) ⊂ L (H ) is called singular if it vanishes on the set K (H ).
Since a singular trace is defined up to trace-class operators C 1 (H ), then by Remark 5 it is obviously not normal.

Dixmier Trace

Recall that only the ideal of trace-class operators has the property that on its positive cone

C 1,+ (H ) := {A ∈ C 1 (H ) : A ≥ 0} the trace norm is linear since A + B 1 = Tr (A + B) = Tr (A) + Tr (B) = A 1 + B 1 for A, B ∈ C 1,+ (H ), see Example 1.
Then, the uniqueness of the trace norm allows to extend the trace to the whole linear space C 1 (H ). Imitation of this idea fails for other symmetricallynormed ideals.

This problem motivates the Dixmier trace construction as a certain limiting procedure involving the • 1,∞ -norm. Let C 1,∞,+ (H ) be a positive cone of the Dixmier ideal. One can try to construct on C 1,∞,+ (H ) a linear, positive and unitarily invariant functional (called trace T ) via extension of the limit (called Lim) of the sequence of properly normalised finite sums of the operator X singular values:

T (X ) := Lim n→∞ 1 1 + ln(n) n j=1 s j (X ) , X ∈ C 1,∞,+ (H ). (3.1)
First, we note that since for any unitary U : H → U , the singular values of X ∈ C ∞ (H ) are invariant: s j (X ) = s j (U X U * ), it is also true for the sequence

σ n (X ) := n j=1 s j (X ) , n ∈ N . (3.2)
Then, the Lim in (3.1) (if it exists) inherits the property of unitarity. Now, we comment that positivity: X ≥ 0 implies the positivity of eigenvalues {λ j (X )} j≥1 and consequently: λ j (X ) = s j (X ). Therefore, σ n (X ) ≥ 0 and the Lim in (3.1) is a positive mapping.

The next problem with the formula for T (X ) is its linearity. To proceed, we recall that if P : H → P(H ) is an orthogonal projection on a finite-dimensional subspace with dim P(H ) = n, then for any bounded operator X ≥ 0 the (3.2) gives σ n (X ) = sup P {Tr H (X P) : dim P(H ) = n} .

(3.3)

As a corollary of (3.3), one obtains the Horn-Ky Fan inequality

σ n (X + Y ) ≤ σ n (X ) + σ n (Y ) , n ∈ N, (3.4)
valid, in particular, for any pair of bounded positive compact operators X and Y . For dim P(H ) ≤ 2n, one similarly gets from (3.3) that

σ 2n (X + Y ) ≥ σ n (X ) + σ n (Y ) , n ∈ N . (3.5)
Motivated by (3.1), we now introduce

T n (X ) := 1 1 + ln(n) σ n (X ) , X ∈ C 1,∞,+ (H ) , (3.6)
valentin.zagrebnov@univ-amu.fr and denote by Lim{T n (X )} n∈N := Lim n→∞ T n (X ) the right-hand side of the functional in (3.1). Note that by (3.6), the inequalities (3.4) and (3.5) yield for n ∈ N

T n (X + Y ) ≤ T n (X ) + T n (Y ) , 1 + ln(2n) 1 + ln(n) T 2n (X + Y ) ≥ T n (X ) + T n (Y ). (3.7)
Since the functional Lim includes the limit n → ∞, the inequalities (3.7) would give a desired linearity of the trace T :

T (X + Y ) = T (X ) + T (Y ) , (3.8)
if one proves that the Lim n→∞ in (3.1) exists and finite for X, Y as well as for X + Y .

To this end, we note that if the right-hand side of (3.1) is finite, then one has (3.8), and hence the Lim{T n (X )} n∈N is a positive linear map Lim : l ∞ (N) → R, which defines a state ω ∈ S (l ∞ (N)) on the Banach space of sequences {T n (•)} n∈N ∈ l ∞ (N). To make this definition more precise, we impose on the state ω the following conditions:

(a) ω(η) ≥ 0 , for ∀η = {η n ≥ 0} n∈N , (b) ω(η) = Lim{η n } n∈N = lim n→∞ η n , if {η n ≥ 0} n∈N is convergent .
By virtue of (a) and (b), the definitions (3.1) and (3.6) imply that for X, Y ∈ C 1,∞,+ (H ), one gets

T (X ) = ω({T n (X )} n∈N ) = lim n→∞ T n (X ) , (3.9)
T (Y ) = ω({T n (Y )} n∈N ) = lim n→∞ T n (Y ) , (3.10)
T (X + Y ) = ω({T n (X + Y )} n∈N ) = lim n→∞ T n (X + Y ) , (3.11)
if the limits in the right-hand sides of (3.9)-(3.11) exist. Now, to ensure (3.8) one has to select ω in such a way that it allows to restore the equality in (3.7), when n → ∞. To this aim, we impose on the state ω the condition of dilation D 2 -invariance.

Let D 2 : l ∞ (N) → l ∞ (N) be dilation mapping η → D 2 (η):

D 2 : (η 1 , η 2 , . . . η k , . . .) → (η 1 , η 1 , η 2 , η 2 , . . . η k , η k , . . .) , ∀η ∈ l ∞ (N). (3.12)
We say that ω is dilation D 2 -invariant if for any η ∈ l ∞ (N) it verifies the property

(c) ω(η) = ω(D 2 (η)). (3.13)
We shall discuss the question of existence, the dilation D 2 -invariant states (the invariant means), on the Banach space l ∞ (N) in Remark 6.

Let X, Y ∈ C 1,∞,+ (H ). Then, applying the property (c) to the sequence η = {ξ 2n := T 2n (X + Y )} ∞ n=1 , we obtain

ω(η) = ω(D 2 (η)) = ω(ξ 2 , ξ 2 , ξ 4 , ξ 4 , ξ 6 , ξ 6 , . . .) . (3.14) Note that if ξ = {ξ n = T n (X + Y )} ∞ n=1
, then the difference of the sequences:

D 2 (η) -ξ = (ξ 2 , ξ 2 , ξ 4 , ξ 4 , ξ 6 , ξ 6 , . . .) -(ξ 1 , ξ 2 , ξ 3 , ξ 4 , ξ 5 , ξ 6 , . . .) , converges to zero if ξ 2n -ξ 2n-1 → 0 as n → ∞.
Then, by virtue of (3.11) and (3.14), this would imply

ω({T 2n (X + Y )} n∈N ) = ω(D 2 ({T 2n (X + Y )} n∈N )) = ω({T n (X + Y )} n∈N ) ,
or by (3.11): lim n→∞ T 2n (X + Y ) = lim n→∞ T n (X + Y ), which by estimates (3.7) would also yield Therefore, to finish the proof of linearity it rests only to check that lim n→∞ (ξ 2nξ 2n-1 ) = 0. To this end, we note that by definitions (3.2) and (3.6), one gets

lim n→∞ T n (X + Y ) = lim n→∞ T n (X ) + lim n→∞ T n (Y ). ( 3
ξ 2n -ξ 2n-1 = 1 ln(2n) - 1 ln(2n -1) σ 2n-1 (X + Y ) + 1 ln(2n) s 2n (X + Y ). (3.16) Since X, Y ∈ C 1,∞,+ (
Tr ω (X ) := Lim n→∞ σ n (X ) 1 + ln(n) , (3.17)
valentin.zagrebnov@univ-amu.fr

where Lim n→∞ is defined by a dilation-invariant state ω ∈ S (l ∞ (N)) on l ∞ (N) that satisfies the properties (a), (b) and (c). Since any self-adjoint operator X ∈ C 1,∞ (H ) has the representation: X = X + -X -, where X ± ∈ C 1,∞,+ (H ), one gets Tr ω (X ) = Tr ω (X + ) -Tr ω (X -). Then, for arbitrary Z ∈ C 1,∞ (H ), the Dixmier trace is Tr ω (Z ) = Tr ω (ReZ ) + iTr ω (Im Z ).

Note that if X ∈ C 1,∞,+ (H ), then definition (3.17) of Tr ω (•) together with definition of the norm • 1,∞ in (2.3) readily imply the estimate Tr ω (X ) ≤ X 1,∞ , which, in turn, yields the inequality for arbitrary Z from the Dixmier ideal C 1,∞ (H ): For more details about different constructions of invariant means and the corresponding Dixmier trace on C 1,∞ (H ), see e.g., [START_REF] Carey | Dixmier traces and some applications in non-commutative geometry[END_REF][START_REF] Lord | Singular Traces. Theory and Applications[END_REF].

|Tr ω (Z )| ≤ Z 1,∞ . ( 3
ω : C 1,∞ (H ) → C is continuous in the • 1,∞ -norm.
Proof (a) Since every operator B ∈ L (H ) is a linear combination of four unitary operators, it is sufficient to prove the equality Tr ω (ZU ) = Tr ω (U Z ) for a unitary operator U and moreover only for Z ∈ C 1,∞,+ (H ). Then, the corresponding equality follows from the unitary invariance:

s j (Z ) = s j (ZU ) = s j (U Z ) = s j (U ZU * ) of singular values of the positive operator Z for all j ≥ 1. (b) Since C ∈ C 1 (H ) yields C 1 < ∞, definition (3.2) implies σ n (C) ≤ C 1 for
any n ≥ 1. Then, by Definition 5 one gets Tr ω (C) = 0. Proof of the last part of the statement is standard. (c) Since the ideal C 1,∞ (H ) is a Banach space and Tr ω : C 1,∞ (H ) → C a linear functional, it is sufficient to consider continuity at X = 0. Then, let the sequence

{X k } k≥1 ⊂ C 1,∞ (H ) converges to X = 0 in • 1,∞ -topology, i.e. by (2.3) lim k→∞ X k 1,∞ = lim k→∞ sup n∈N 1 1 + ln(n) σ n (X k ) = 0 . (3.19) Since (3.18) implies |Tr ω (X k )| ≤ X k 1,∞ , the assertion follows from (3.19).
Therefore, the Dixmier construction gives an example of a singular trace in the sense of Definition 4.

Trotter-Kato Product Formulae in the Dixmier Ideal

Let A ≥ 0 and B ≥ 0 be two non-negative self-adjoint operators in a separable Hilbert space H and let the subspace H 0 := dom(A 1/2 ) ∩ dom(B 1/2 ). It may happen that dom(A) ∩ dom(B) = {0}, but the form-sum of these operators:

H = A . + B is well-defined in the subspace H 0 ⊆ H .
Kato proved in [START_REF] Kato | Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroups[END_REF] that under these conditions the Trotter product formula s -lim n→∞ e -t A/n e -t B/n n = e -t H P 0 , t ≥ 0, (

converges in the strong operator topology uniformly in t ∈ [0, T ], 0 < T < +∞ , where P 0 denotes the orthogonal projection from H onto H 0 . Moreover, in [START_REF] Kato | Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroups[END_REF] it was also shown that the product formula is true not only for the exponential function e -x , x ≥ 0, but for a whole class of Borel measurable functions f (•) and g(•), which are defined on R + 0 := [0, ∞) and satisfy the conditions:

0 ≤ f (x) ≤ 1, f (0) = 1, f ′ (+0) = -1, (4.2) 0 ≤ g(x) ≤ 1, g(0) = 1, g ′ (+0) = -1. (4.3)
Namely, the main result of [START_REF] Kato | Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroups[END_REF] says that besides (4.1) one also gets convergence

τ -lim n→∞ ( f (t A/n)g(t B/n)) n = e -t H P 0 , t ≥ 0, (4.4)
uniformly in t ∈ [0, T ], 0 < T < +∞, if topology τ = s. Product formulae of the type (4.4) are called the Trotter-Kato product formulae for functions (4.2), (4.3), which are called the Kato functions K . Note that K is closed with respect to the products of Kato functions.

For some particular classes of the Kato functions we refer to [START_REF] Neidhardt | On error estimates for the Trotter-Kato product formula[END_REF][START_REF] Zagrebnov | Topics in the Theory of Gibbs Semigroups[END_REF]. In the following, it is useful to consider instead of f (x)g(x) two Kato functions: g(x/2) f (x)g(x/2) and f (x/2)g(x) f (x/2) that produce the self-adjoint operator families F(t) := g(t B/2) f (t A)g(t B/2) and T (t) := f (t A/2)g(t B) f (t A/2), t ≥ 0.

(4.5) Since [START_REF] Neidhardt | The Trotter product formula for Gibbs semigroup[END_REF] it is known that the lifting of the topology of convergence in (4.4) to the operator norm τ = • needs more conditions on operators A and B as well as on the key Kato functions f, g ∈ K . One finds a discussion and more references on this subject in [START_REF] Zagrebnov | Topics in the Theory of Gibbs Semigroups[END_REF]. Here, we quote a result that will be used below for the Trotter-Kato product formulae in the Dixmier ideal C 1,∞ (H ).

Consider the class K β of Kato functions, which is defined in [START_REF] Ichinose | The norm convergence of the Trotter-Kato product formula with error bound[END_REF][START_REF] Ichinose | Note on the paper "the norm convergence of the Trotter-Kato product formula with error bound" by Ichinose and Tamura[END_REF] as given below:

(i) Measurable functions 0 ≤ h ≤ 1 on R + 0 , such that h(0) = 1 and h ′ (+0) = -1. (ii) For ε > 0 there exists δ = δ(ε) < 1, such that h(s) ≤ 1 -δ(ε) for s ≥ ε and

[h] β := sup s>0 |h(s) -1 + s| s β < ∞ , for 1 < β ≤ 2 .
The standard examples are h(s) = e -s and h(s) = (1 + a -1 s) -a , a > 0.

Below we consider the class K β and a particular case of generators A and B, such that for the Trotter-Kato product formulae the estimate of the convergence rate is optimal. Proposition 4.1 ([11]) Let f, g ∈ K β with β = 2, and let A, B be non-negative selfadjoint operators in H such that the operator sum C := A + B is self-adjoint on domain dom(C) := dom(A) ∩ dom(B). Then, the Trotter-Kato product formulae converge for n → ∞ in the operator norm:

[ f (t A/n)g(t B/n)] n -e -tC = O(n -1 ) , [g(t B/n) f (t A/n)] n -e -tC = O(n -1 ) , F(t/n) n -e -tC = O(n -1 ) , T (t/n) n -e -tC = O(n -1 ).
Note that corresponding to each formula, error bounds O(n -1 ) are equal up to coefficients {Γ j > 0} 4 j=1 and that each rate of convergence

Γ j ε(n) = O(n -1 ), j = 1, . . . 4, is optimal.
The first lifting lemma yields sufficient conditions that allow to strengthen the strong operator convergence to the • φ -norm convergence in the symmetricallynormed ideal C φ (H ). Since C φ (H ) is a Banach space with symmetric norm (1.13) that verifies Z X φ ≤ Z X φ , one gets the estimate

(Z (t/r ) -Z )Y X φ ≤ (Z (t/r ) -Z )Y X φ , (4.9)
which together with (4.8) give the proof of (4.7).

The second lifting lemma allows to estimate the rate of convergence of the Trotter-Kato product formula in the norm (1.19) of symmetrically-normed ideal C φ (H ) via the error bound ε(n) in the operator norm due to Proposition 4.1.

Lemma 4.3 Let A and B be non-negative self-adjoint operators on the separable

Hilbert space H that satisfy the conditions of Proposition 4.1. Let f, g ∈ K 2 be such that F(t 0 ) ∈ C φ (H ) for some t 0 > 0.

If Γ t 0 ε(n), n ∈ N is the operator-norm error bound away from t 0 > 0 of the Trotter-Kato product formula for { f (t A)g(t B)} t≥0 , then for some Γ φ 2t 0 > 0 the function ε φ (n) := {ε([n/2]) + ε([(n + 1)/2])}, n ∈ N defines the error bound away from 2t 0 of the Trotter-Kato product formula in the ideal C φ (H ):

[ f (t A/n)g(t B/n)] n -e -tC φ = Γ φ 2t 0 ε φ (n) , n → ∞. t ≥ 2t 0 . ( 4
( f (t A/n)g(t B/n)) n -e -tC = (4.11) ( f (t A/n)g(t B/n)) k -e -ktC/n ( f (t A/n)g(t B/n)) m + e -ktC/n ( f (t A/n)g(t B/n)) m -e -mtC/n .
Since by conditions of lemma F(t 0 ) ∈ C φ (H ), definition (4.5) and representation

f (t A/n)g(t B/n)) m = f (t A/n)g(t B/n) 1/2 F(t/n) m-1 g(t B) 1/2 yield ( f (t A/n)g(t B/n)) m φ ≤ F(t 0 ) φ , (4.12)
for t such that t 0 ≤ (m -1)t/n ≤ (m -1)t 0 and m -1 ≥ 1.

Note that for self-adjoint operators e -tC and F(t) by Araki's log-order inequality for compact operators [START_REF] Araki | On an inequality of Lieb and Thirring[END_REF], one gets for kt/n ≥ t 0 the bound of e -ktC/n in the • φnorm:

e -ktC/n φ ≤ F(t 0 ) φ . (4.13)

Since by Definitions 2 and 3 the ideal C φ (H ) is a Banach space, and from (4.11)-(4.13), we obtain the estimate

( f (t A/n)g(t B/n)) n -e -tC φ ≤ (4.14) F(t 0 ) φ ( f (t A/n)g(t B/n)) k -e -ktC/n + F(t 0 ) φ ( f (t A/n)g(t B/n)) m -e -mtC/n , for t such that (1 + (k + 1)/(m -1))t 0 ≤ t ≤ nt 0 , m ≥ 2 and t ≥ (1 + m/k)t 0 .
Now, by conditions of lemma Γ t 0 ε(•) is the operator-norm error bound away from t 0 , for any interval [a, b] ⊆ (t 0 , +∞). Then, there exists n 0 ∈ N such that

( f (t A/n)g(t B/n)) k -e -ktC/n ≤ Γ t 0 ε(k) , (4.15)
for kt/n ∈ [a, b] ⇔ t ∈ [(1 + m/k)a, (1 + m/k)b] and ( f (t A/n)g(t B/n)) m -e -mtC/n ≤ Γ t 0 ε(m) , (4.16
[τ 0 , τ ] ⊆ (2t 0 , +∞), we find that [τ 0 , τ ] ⊆ [(1 + (k + 1)/ (m -1))t 0 , nt 0 ] for sufficiently large n. Moreover, choosing [τ 0 /2, τ/2] ⊆ (a, b) ⊆ (t 0 , +∞), we satisfy [τ 0 , τ ] ⊆ [(1 + m/k)a, (1 + m/k)b] and [τ 0 , τ ] ⊆ [(1 + k/ m)a, (1 + k/m)b] again for sufficiently large n.
Thus, for any interval [τ 0 , τ ] ⊆ (2t 0 , +∞) there is n 0 ∈ N such that (4.14)-(4.16) hold for t ∈ [τ 0 , τ ] and n ≥ n 0 . Therefore, (4.14) yields the estimate

( f (t A/n)g(t B/n)) n -e -tC φ ≤ (4.17) Γ t 0 F(t 0 ) φ {ε([n/2]) + ε([(n + 1)/2])} , for t ∈ [τ 0 , τ ] ⊆ (2t 0 , +∞) and n ≥ n 0 . Hence, Γ φ 2t 0 := Γ t 0 F(t 0 ) φ and Γ φ 2t 0 ε φ (•)
is an error bound in the Trotter-Kato product formula (4.10) away from 2t 0 in C φ (H ) for the family { f (t A)g(t B)} t≥0 .

The lifting Lemma 4.2 allows to extend the proofs for other approximants: {g(t B) f (t A)} t≥0 , {F(t)} t≥0 and {T (t)} t≥0 .

valentin.zagrebnov@univ-amu.fr Now, we apply Lemma 4.3 in Dixmier ideal C φ (H ) = C 1,∞ (H ). This concerns the norm convergence (4.10), but also the estimate of the convergence rate for Dixmier traces:

|Tr ω (e Note that for the particular case of Proposition 4.1, these arguments yield for (4.17) the explicit convergence rate asymptotics O(n -1 ) for the Trotter-Kato formulae and consequently, the same asymptotics for convergence rates of the Trotter-Kato formulae for the Dixmier trace (4.18), (4.19).

Therefore, we proved in the Dixmier ideal C 1,∞ (H ) the following assertion. away from 2t 0 . The rate O(n -1 ) of convergence is optimal in the sense of [START_REF] Ichinose | Note on the paper "the norm convergence of the Trotter-Kato product formula with error bound" by Ichinose and Tamura[END_REF]. By virtue of (4.19), the same asymptotics O(n -1 ) of the convergence rate are valid for convergence of the Trotter-Kato formulae for the Dixmier trace: Optimality of the estimates in Theorem 1 is a heritage of the optimality in Proposition 4.1. Recall that, in particular, this means that in contrast to the Lie product formula for valentin.zagrebnov@univ-amu.fr bounded generators A and B, the symmetrisation of approximants { f (t)g(t)} t≥0 , and {g(t) f (t)} t≥0 by {F(t)} t≥0 and {T (t)} t≥0 , does not yield (in general) the improvement of the convergence rate, see [START_REF] Ichinose | Note on the paper "the norm convergence of the Trotter-Kato product formula with error bound" by Ichinose and Tamura[END_REF][START_REF] Tamura | A remark on operator-norm convergence of Trotter-Kato product formula[END_REF] and discussion in [START_REF] Zagrebnov | Trotter-Kato product formula: some recent results[END_REF].

We resume that the lifting Lemma 4.2 and 4.3 are a general method to study the convergence in symmetrically-normed ideals C φ (H ) as soon as it is established in L (H ) in the operator-norm topology. The crucial is to check that for any of the key Kato functions (e.g. for {F(t)} t≥0 ) there exists t 0 > 0 such that F(t)| t≥t 0 ∈ C φ (H ). Sufficient conditions for that one can be found in [START_REF] Neidhardt | Fractional powers of self-adjoint operators and Trotter-Kato product formula[END_REF][START_REF] Neidhardt | Trotter-Kato product formula and operator-norm convergence[END_REF][START_REF] Neidhardt | On the operator-norm convergence of the Trotter-Kato product formula[END_REF] or in [START_REF] Zagrebnov | Topics in the Theory of Gibbs Semigroups[END_REF].

. 8 )

 8 Note that the left-and right-hand sides of(1.8) are the simplest examples of symmetric norming functions on domain c + f :φ ∞ (ξ ) := ξ 1 and φ 1 (ξ ) := ∞ j=1 ξ j .(1.9)By Definition 1, the observations (1.8) and (1.9) yieldφ ∞ (ξ ) := max j≥1 |ξ j | , φ 1 (ξ ) := ∞ j=1 |ξ j | ,(1.10)φ ∞ (ξ ) ≤ φ(ξ ) ≤ φ 1 (ξ ) , for all ξ ∈ c f .We denote by ξ * := {ξ * 1 , ξ * 2 , . . . } a decreasing rearrangement:ξ * 1 = sup j≥1 |ξ j | , ξ * 1 + ξ * 2 = sup i = j {|ξ i | + |ξ j |}, . . . ,of the sequence of absolute values {|ξ n |} n≥1 , i.e. ξ * 1 ≥ ξ * 2 ≥ . . . . Then, ξ ∈ c f implies ξ * ∈ c f and by (1.5) one obtains also that φ(ξ ) = φ(ξ * ), ξ ∈ c f . (1.11)

Remark 5

 5 The canonical trace Tr H (•) is nontrivial only on domain, which is the trace-class ideal C 1 (H ), see Example 1. We recall that it is characterised by the property of normality: Tr H (sup α B α ) = sup α Tr H (B α ), for every directed increasing bounded family {B α } α∈ of positive operators from C 1,+ (H ).

. 15 )

 15 Now, summarising (3.9), (3.10),(3.11) and(3.15), we obtain the linearity (3.8) of the limiting functional T on the positive cone C 1,∞,+ (H ) if it is defined by the corresponding D 2 -invariant state ω or a dilation-invariant mean.

Definition 5

 5 H ), we obtain that lim n→∞ s 2n (X + Y ) = 0 and that σ 2n-1 (X + Y ) = O(ln(2n -1)). Then taking into account that (1/ln(2n) -1/ ln(2n -1)) = o(1/ln(2n -1)) one gets that for n → ∞ the right-hand side of (3.16) converges to zero.To conclude our construction of the trace T (•) we note that by linearity (3.8) one can uniquely extend this functional from the positive cone C 1,∞,+ (H ) to the real subspace of the Banach space C 1,∞ (H ), and finally to the entire ideal C 1,∞ (H ). The Dixmier trace Tr ω (X ) of the operator X ∈ C 1,∞,+ (H ) is the value of the linear functional (3.1):

. 18 )

 18 Remark 6 A decisive for construction of the Dixmier trace Tr ω (•) is the existence of the invariant mean ω ∈ S (l ∞ (N)) ⊂ (l ∞ (N)) * . Here, the space (l ∞ (N)) * is dual to the Banach space of bounded sequences. Then, by the Banach-Alaoglu theorem, the convex set of states S (l∞ (N)) is compact in (l ∞ (N)) * in the weak* topology. Now, for any φ ∈ S (l ∞ (N)), the relation φ(D 2 (•)) =: (D * 2 φ)(•) defines the dual D * 2 -dilation on the set of states. By definition (3.12), this map is such that D * 2 : S (l ∞ (N)) → S (l ∞ (N)), as well as continuous and affine (in fact, linear). Then, by the Markov-Kakutani theorem, the dilation D * 2 has a fix point ω ∈ S (l ∞ (N)) : D * 2 ω = ω. This observation justifies the existence of the invariant mean (c) for D 2dilation. Note that Remark 6 has a straightforward extension to any D k -dilation for k > 2, which is defined similar to (3.12). Since dilations for different k ≥ 2 commute, the extension of the Markov-Kakutani theorem yields that the commutative family F = {D * k } k≥2 has in S (l ∞ (N)) the common fix point ω = D * 2 ω. Therefore, Definition 5 of the Dixmier trace does not depend on the degree k ≥ 2 of dilation D k .

Proposition 3 . 1

 31 The Dixmier trace has the following properties: (a) For any bounded operator B ∈ L (H ) and Z ∈ C 1,∞ (H ) one has Tr ω (Z B) = Tr ω (B Z ). (b) Tr ω (C) = 0 for any operator C ∈ C 1 (H ) from the trace-class ideal, which is the closure of finite-rank operators K (H ) for the • 1 -norm. (c) The Dixmier trace Tr

Lemma 4 . 2

 42 Let self-adjoint operators: X ∈C φ (H ), Y ∈ C ∞ (H ) and Z ∈ L (H ). If {Z (t)} t≥0 is a family of self-adjoint bounded operators such that s -lim t→+0 Z (t) = Z , (4.6) t/r ) -Z )Y X φ = lim r →∞ sup t∈[0,τ ] X Y (Z (t/r ) -Z ) φ = 0 , (4.7)for any τ ∈ (0, ∞).Proof Note that (4.6) yields the strong operator convergence s -lim r →∞ Z (t/r ) = Z , uniformly in t ∈ [0, τ ]. Since Y ∈ C ∞ (H ), this implies lim r →∞ sup t∈[0,τ ] (Z (t/r ) -Z )Y = 0 . (4.8)

. 10 )

 10 Here [x] := max{l ∈ N 0 : l ≤ x}, for x ∈ R + 0 .Proof To prove the assertion for the family { f (t A)g(t B)} t≥0 , we use decompositions n = k + m, k ∈ N and m = 2, 3, . . . , n ≥ 3, for representation

  ) for mt/n ∈ [a, b] ⇔ t ∈ [(1 + k/m)a, (1 + k/m)b] for all n > n 0 . Setting m := [(n + 1)/2] and k = [n/2], n ≥ 3, we satisfy n = k + m and m ≥ 2, as well as, lim n→∞ (k + 1)/(m -1) = 1, lim n→∞ m/k = 1 and lim n→∞ k/m = 1. Hence, for any interval

Theorem 1

 1 Let f, g ∈ K β with β = 2, and let A, B be non-negative self-adjoint operators in H such that the operator sum C := A + B is self-adjoint on domain dom(C) := dom(A) ∩ dom(B).If F(t 0 ) ∈ C 1,∞ (H ) for some t 0 > 0, then the Trotter-Kato product formulae converge for n → ∞ in the • 1,∞ -norm:[ f (t A/n)g(t B/n)] ne -tC 1,∞ = O(n -1 ) , [g(t B/n) f (t A/n)] ne -tC 1,∞ = O(n -1 ) , F(t/n) ne -tC 1,∞ = O(n -1 ) , T (t/n) ne -tC 1,∞ = O(n -1 ) ,

  |Tr ω ([ f (t A/n)g(t B/n)] n ) -Tr ω (e -tC )| = O(n -1 ) , |Tr ω (F(t/n) n ) -Tr ω (e -tC )| = O(n -1 ) , |Tr ω (T (t/n) n ) -Tr ω (e -tC )| = O(n -1 ) ,away from 2t 0 .

  Remark 1 By virtue of inequality (1.7) and by definition of symmetric norm (1.19), the so-called dominance property holds: if X ∈ C φ (H ), Y ∈ C ∞ (H ) and

	n	n
	s j (Y ) ≤	s j (X ) ,	n = 1, 2, . . . ,
	j=1	j=1
	then Y ∈ C φ (H ) and Y φ ≤ X φ .
	Remark 2 To distinguish in (1.20) nontrivial ideals C φ one needs some criteria based
	on the properties of φ or • φ . For example, any symmetric norming function (1.11)
	defined by

  ) one can also describe the space C 1 (H ) * , which is a predual of C 1 (H ), i.e. its dual (C 1 (H ) * ) * = C 1 (H ). To this aim for each fixed B ∈ C 1 (H ), we consider the functionals A → Tr H (AB) on L (H ). It is known that they are not all continuous linear functional on bounded operators L (H ), i.e. C 1 (H ) ⊂ L (H ) Now, we note that under duality relation (2.6), the Dixmier ideal C 1,∞ (H ) is the dual of the Macaev

* , but they yield the entire dual only of compact operators, i.e.

C 1 (H ) = C ∞ (H ) * . Hence, C 1 (H ) * = C ∞ (H ).

valentin.zagrebnov@univ-amu.fr

  -tC ) -Tr ω (F(t/n) n )| ≤ Γ ω ε ω (n). (4.18)In fact, it is the same (up to Γ ω ) for all Trotter-Kato approximants:{T (t)} t≥0 , { f (t)g(t)} t≥0 and {g(t) f (t)} t≥0 .Indeed, since by inequality (3.18) and Lemma 4.3 for t ∈ [τ 0 , τ ] and n ≥ n 0 , one has|Tr ω (e -tC ) -Tr ω (F(t/n) n )| ≤ e -tC -F(t/n) n 1,∞ ≤ Γ φ 2t 0 ε 1,∞ (n) ,(4.19)we obtain for the rate in (4.18):ε ω (•) = ε 1,∞ (•).Therefore, the estimate of the convergence rate for Dixmier traces (4.18) and for • 1,∞ -convergence in (4.19) are entirely defined by the operator-norm error bound ε(•) from Lemma 4.3 and have the following form:

ε 1,∞ (n) := {ε([n/2]) + ε([(n + 1)/2])} , n ∈ N . (

4

.20)

valentin.zagrebnov@univ-amu.fr

Acknowledgements I am thankful to referee for useful remarks and suggestions.


Kristel Carolina

Meza Fajardo

Hideo Aochi

Apostolos S Papageorgiou

Kristel C Meza-Fajardo

Observed Surface Wave Energy in the Los Angeles Basin Induced By The July 6th, 2019 Mw 7.1 Ridgecrest Earthquake

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The dominant frequency of the most energetic Love waves is < 0.15 Hz. Stations that have recorded very strong Love waves are located in the vicinity of the deepest parts of the basin and in close proximity to regions where the basin displays the greatest gradients (south of downtown LA and reasonably close to the LA airport). Low frequency Love wave energy is observed also at the south-eastern part of the basin where its depth displays much smaller gradients, but it is not as strong. The duration of Love waves increases as one moves to the deepest parts of the basin. However, the increase in the duration never exceeds 10% of the duration of the remaining energy present in the signal.

The dominant frequency of Rayleigh wave energy lies in the range 0.10 to 0.20 Hz, consistently throughout the basin. The Rayleigh waves are less energetic than Love waves and their peak amplitudes are observed more to the south. Rayleigh waves have been identified also outside the basin. The distributions of their duration is less clear as compared to those of Love waves. Nevertheless, the largest delay time is observed in the vicinity of the SW edge of the basin, and the duration does not exceed that of the remaining signal (i.e. after extraction of the Rayleigh wave energy).


Hagen Neidhardt
email: hagen.neidhardt@wias-berlin.de

Artur Stephan
email: stephan@math.hu-berlin.de

Valentin A Zagrebnov Dedicated

Haim Brezis

Louis Nirenberg

V A Zagrebnov
email: valentin.zagrebnov@univ-amu.fr

Operator-norm convergence of the Trotter product formula on Hilbert and Banach spaces: a short survey

We give a review of results on the operator-norm convergence of the Trotter product formula on Hilbert and Banach spaces, which is focused on the problem of its convergence rates. Some recent results concerning evolution semigroups are presented in details.

Introduction

Recall that the product formula e -τC = lim n→∞ e -τA/n e -τB/n n , τ ≥ 0, (

was established by S. Lie (in 1875) for matrices where C := A + B. The proof of formula (1.1) can be carried over easily to bounded operators on Banach spaces. Moreover, a straightforward computation shows that the convergence rate is O(1/n), i.e. sup τ∈[0,T ] e -τA/n e -τB/ne -τC/n = O(1/n).

(1.2) H. Trotter [START_REF] Trotter | On the product of semi-groups of operators[END_REF] has extended this result to unbounded operators A and B on Banach spaces, but in the strong operator topology. He proved that if A and B are generators of contractions semigroups on a separable Banach space such that the algebraic sum A + B is a densely defined closable operator and the closure C = A + B is a generator of a contraction semigroup, then e -τC = s-lim n→∞ e -τA/n e -τB/n n , (

uniformly in τ ∈ [0, T ] for any T > 0. Formula (1.3) is often called the Trotter or the Lie-Trotter product formula. It was a long-time belief that this formula is valid only in the strong operator topology. But in nineties it was discovered that under certain quite standard assumptions the strong convergence of the Trotter product formula can be improved to the operator-norm convergence. In the following we give a review of these results.

The paper is organised as follows. In Section 2.1 we give an overview on operator-norm convergence of the Trotter product formula if the generators A and B are non-negative self-adjoint operators. Section 2.2 summarises the case when one of the generator is only a maximal accretive operator. Section 2.3 is devoted to the evolution case, which arises in the theory of the abstract non-autonomous Cauchy problem. These results are commented in Section 2. [START_REF] Cachia | Zagrebnov Operator-norm convergence of the Trotter product formula for sectorial generators[END_REF].

Section 3 is concerned with the operator-norm convergence of the Trotter product formula on the Banach spaces. Section 3.1 presents the results under the assumption that one of generators is for holomorphic semigroup. Section 3.2 considers again the evolution case but on Banach spaces. The relation between evolution semigroups and propagators is explained in Section 3.3. We comment the results in Section 3. [START_REF] Cachia | Zagrebnov Operator-norm convergence of the Trotter product formula for sectorial generators[END_REF].

In Section 4 we collect some examples and counterexamples. They show what is expectable and what is not and even surprising.

We use below the following notations and definitions.

1. We use a definition of the semigroup generator C (1.3), which differs from the standard one by a minus, as it is in the book [START_REF] Kato | Perturbation theory for linear operators[END_REF]. 2. Furthermore, we widely use the so-called Landau symbols:

g(n) = O( f (n)) ⇐⇒ lim sup n→∞ g(n) f (n) < ∞ , g(n) = o( f (n)) ⇐⇒ lim sup n→∞ g(n) f (n) = 0 , g(n) = Θ ( f (n)) ⇐⇒ 0 < lim inf n→∞ g(n) f (n) ≤ lim sup n→∞ g(n) f (n) < ∞ , g(n) = ω( f (n)) ⇐⇒ lim sup n→∞ g(n) f (n) = ∞ .
3. We use the notation C 0,β ([0, T ]) for the Hölder (β ∈ (0, 1)) and, respectively, for the Lipschitz (β = 1) continuous functions.

2 Trotter product formula on Hilbert spaces

Self-adjoint case

Considering the Trotter product formula on a separable Hilbert space H T. Kato has shown in [START_REF] Kato | On the Trotter-Lie product formula[END_REF][START_REF] Kato | Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroups[END_REF] that for non-negative operators A and B the Trotter formula (1.3) holds in the strong operator topology if dom( √ A) ∩ dom( √ B) is dense in the Hilbert space and C = A +B is the form-sum of operators A and B.

Naturally the problem arises whether Kato's result can be extended to the operator-norm convergence. A first attempt in this direction was undertaken by Rogava [START_REF] Dzh | Error bounds for Trotter-type formulas for self-adjoint operators[END_REF]. He claimed that if A and B are non-negative self-adjoint operators such that dom(A) ⊆ dom(B) and the operator-sum:

C = A + B, is self-adjoint, then (e -τA/n e -τB/n ) n -e -τC = O(ln(n)/ √ n), n → ∞, (2.4)
holds. In [START_REF] Neidhardt | On error estimates for the Trotter-Kato product formula[END_REF] it was shown that if one substitutes in above conditions the selfadjointness of the operator-sum by the A-smallness of B with a relative bound less then one, then (2.4) is true with the rate of convergence improved to

(e -τA/n e -τB/n ) n -e -τC = O(ln(n)/n), n → ∞.
The problem in its original formulation was finally solved in [START_REF] Ichinose | Note on the paper: "The norm convergence of the Trotter-Kato product formula with error bound[END_REF]. There it was shown that the best possible in this general setup rate (1.2) holds if the operator sum: C = A + B, is already a self-adjoint operator. Obviously, Rogava's result, as well as many other results (including [START_REF] Neidhardt | On error estimates for the Trotter-Kato product formula[END_REF]), when the operator sum of generators is self-adjoint, follow from [START_REF] Ichinose | Note on the paper: "The norm convergence of the Trotter-Kato product formula with error bound[END_REF] .

A new direction comes due to results for the fractional-power conditions. In [START_REF] Neidhardt | Fractional powers of self-adjoint operators and Trotter-Kato product formula[END_REF], with elucidation in [START_REF] Ichinose | Trotter-Kato product formula and fractional powers of self-adjoint generators[END_REF], it was proven that assuming

dom(C α ) ⊆ dom(A α ) ∩ dom(B α ), α ∈ (1/2, 1), C = A +B, (2.5) and dom(A 1/2 ) ⊆ dom(B 1/2 ) (2.6) one obtains that sup τ∈[0,T ] (e -τA/n e -τB/n ) n -e -τC = O(n -(2α-1) ).
Notice that formally α = 1 yields the rate obtained in [START_REF] Ichinose | Note on the paper: "The norm convergence of the Trotter-Kato product formula with error bound[END_REF].

We remark also that the results of [START_REF] Ichinose | Trotter-Kato product formula and fractional powers of self-adjoint generators[END_REF][START_REF] Neidhardt | Fractional powers of self-adjoint operators and Trotter-Kato product formula[END_REF] do not cover the case α = 1/2. Although, it turns out that in this case the Trotter product formula converges on the operator norm: sup

τ∈[0,T ] (e -τA/n e -τB/n ) n -e -τC = o(1) , if √ B is relatively compact with respect to √ A, i.e. √ B(I + A) -1/2 is compact, see [16].

Nonself-adjoint case

Another direction was related with attempts to extend the the Trotter, and the Trotter-Kato, product formulae to the case of nonself-adjoint sectorial generators [START_REF] Cachia | Operator-norm approximation of semigroups by quasisectorial contractions[END_REF]. Let A be a non-negative self-adjoint operator and let B be a maximal accretive (ℜe(B f , f ) ≥ 0 for f ∈ dom(B)) operator, such that dom(A) ⊆ dom(B) and dom(A) ⊆ dom(B * ).

If B is A-small with a relative bound less than one, then the rate estimate (2.4) holds, for generator C, which is the well-defined maximal accretive operator-sum: C = A + B, see [START_REF] Cachia | Accretive perturbations and error estimates for the Trotter product formula[END_REF].

In [START_REF] Neidhardt | Comments on the Trotter product formula error-bound estimates for nonself-adjoint semigroups[END_REF] this result was generalised as follows. Let A be a non-negative self-adjoint operator and let B be a maximal accretive operator such that dom(A) ⊆ dom(B) and B is A-small with relative bound less than one. If the condition

dom((C * ) α ) ⊆ dom(A α ) ∩ dom((B * ) α ), C = A + B ,
is satisfied for some α ∈ (0, 1], then the norm-convergent Trotter product formula:

sup τ∈[0,T ] (e -τA/n e -τB/n ) n -e -τC = O(ln(n)/n α ) , holds as n → ∞.
In fact, more results are known about the operator-norm Trotter product formula convergence for nonself-adjoint semigroups, but without the rate estimates, see [START_REF] Cachia | Zagrebnov Operator-norm convergence of the Trotter product formula for sectorial generators[END_REF].

Evolution case

At the first glance a very different result about a Trotter-type product formula was obtained in [START_REF] Ichinose | Error estimate in operator norm of exponential product formulas for propagators of parabolic evolution equations[END_REF]. The authors consider instead of the self-adjoint operator B a family {B(t)} t∈[0,T ] of self-adjoint operators on the separable Hilbert space H such that the condition dom

(A α ) ⊆ dom(B(t)), t ∈ [0, T ], (2.7)
is satisfied for some α ∈ [0, 1) for A, which is a non-negative self-adjoint operator.

Then the operator sum

C(t) = A + B(t)
∂ t u(t) = -C(t)u(t), t ∈ [0, T ] , (2.8)
corresponding to the non-autonomous Cauchy problem with initial condition u 0 = u(0) for t = 0. It turns out that equation (2.8) admits a propagator {U(t, s)} (t,s)∈ ∆ , ∆ = {(t, s) ∈ [0, T ] × [0, T ] : 0 ≤ s ≤ t ≤ T }, which solves this problem. We remind that a family {U(t, s)} (t,s)∈∆ of bounded operators is called a propagator if the operator-valued function U(•, •) : ∆ -→ B(H) is strongly continuous and verifies the conditions:

U(t,t) = I for t ∈ [0, T ],
(2.9)

U(t, s) = U(t, r)U(r, s) for t, r, s ∈ [0, T ] with s ≤ r ≤ t . (2.10)
Let {t j } N j=0 be a partition of the closed interval [0,t]:

0 = t 0 < t 1 < . . . < t N-1 < t N = t, t j = τ j, τ = t/N , for any 0 < t ≤ T . Further, let Q j (t, s; n) = e -t-s n A e -t-s n B(s+ j t-s n ) , j = 0, 1, . . . , n , (2.11)
and

E(t, s; n) := Q n-1 (t, s; n)Q n-2 (t, s; n) × • • • × Q 1 (t, s; n)Q 0 (t, s; n) := (n-1)← ∏ j=0 Q j (t, s ; n) , (2.12)
where the symbol ∏ n← j=1 means that the product is increasingly ordered in j from the right to the left. If in addition to assumption (2.7), the condition

A -α (B(t) -B(s))A -α ≤ L 1 |t -s|, t, s ∈ [0, T ], L 1 > 0, ( 2.13)
is satisfied, then in [START_REF] Ichinose | Error estimate in operator norm of exponential product formulas for propagators of parabolic evolution equations[END_REF] it was proved that the propagator {U(t, s)} (t,s)∈ ∆ , which solves the Cauchy problem (2.8), admits the approximation

sup t∈[0,T ] U(t, 0) -E n (t, 0; n) = O(ln(n)/n) as n → ∞. (2.14)
Scrutinising the proof in [START_REF] Ichinose | Error estimate in operator norm of exponential product formulas for propagators of parabolic evolution equations[END_REF] one finds that in fact the claim (2.14) can be slightly generalised to any interval ∆ sup

(t,s)∈ ∆ U(t, s) -E n (t, s; n) = O(ln(n)/n) as n → ∞. (2.15)
At the first glance, it seems that the result (2.15) is quite far from the Trotter product formula. However, this is not the case. To show this we follow the evolution semigroup approach to evolution equations developed in [START_REF] Neidhardt | Convergence rate estimates for Trotter product approximations of solution operators for non-autonomous Cauchy problems[END_REF][START_REF] Neidhardt | Linear non-autonomous Cauchy problems and evolution semigroups[END_REF]. Let us introduce the Hilbert space L 2 ([0, T ], H) and consider on this space the semigroup

(U (τ) f )(t) := U(t,t -τ)χ [0,T ] (t -τ) f (t -τ), f ∈ L 2 ([0, T ], H), (2.16) t ∈ [0, T ]. It turns out that {U (τ)} τ∈R + is a C 0 -semigroup on the Hilbert space of the H-valued trajectories L 2 ([0, T ], H)
. By K we denote its generator, then U (τ) = e -τK , τ ≥ 0.

Let us introduce two multiplication operators

(A f )(t) = A f (t), f ∈ dom(A ), dom(A ) := f ∈ L 2 ([0, T ], H) : f (t) ∈ dom(A) for a.e. t ∈ [0, T ] A f (t) ∈ L 2 ([0, T ], H) , (2.17)
and

(B f )(t) = B(t) f (t), f ∈ dom(B), dom(B) := f ∈ L 2 ([0, T ], H) : f (t) ∈ dom(B(t)) for a.e. t ∈ [0, T ] B(t) f (t) ∈ L 2 ([0, T ], H) . (2.18)
Note that both operators are well-defined and self-adjoint in L 2 ([0, T ], H). Moreover, condition (2.7) yields: dom(A ) ⊆ dom(B).

By D 0 we define in L 2 ([0, T ], H) the generator of the right-shift semigroup e -τD 0 given by (e

-τD 0 f )(t) = χ [0,T ] (t -τ) f (t -τ), f ∈ L 2 ([0, T ], H).
Notice that the operator D 0 is defined by

(D 0 f )(t) = ∂ ∂t f (t), f ∈ dom(D 0 ) := { f ∈ W 2,2 ([0, T ], H) : f (0) = 0}.
Collecting these definitions we introduce the operator

( K f ) = D 0 f + A f + B f , f ∈ dom( K ) := dom(D 0 ) ∩ dom(A ) ∩ dom(B). ( 2
(K 0 f )(t) = D 0 f + A f , f ∈ dom(K 0 ) := dom(D 0 ) ∩ dom(A ).
Then operator K 0 is a generator of the C 0 -semigroup, which has the form

(e -τK 0 f )(t) = e -τA χ t∈[0,T ] (t -τ) f (t -τ), f ∈ L 2 ([0, T ], H).
Note that we obviously get that

K = K 0 + B = K 0 + B.
For the pair {K 0 , B} one obtains the following result.

(e -τK 0 /n e -τB/n ) n -e -τK = O(ln(n)/n) for n → ∞ , (2.20
)

and sup τ≥0 (e -τB/n e -τK 0 /n ) n -e -τK = O(ln(n)/n) for n → ∞ . (2.

21)

Proof. A straightforward computation shows that for f ∈ L 2 ([0, T ], H) one gets

((e -τK 0 /n e -τB/n ) n f )(t) = E(t,t -τ ; n)χ [0,T ] (t -τ) f (t -τ),
where E(t, s ; n) is given by (2.12). Taking into account (2.16) we obtain

((e -τK 0 /n e -τB/n ) n f )(t) -(e -τK f )(t) = (E(t,t -τ ; n) -U(t,t -τ))χ [0,T ] (t -τ) f (t -τ) ,
which yields the estimate

((e -τK 0 /n e -τB/n ) n f )(t) -(e -τK f )(t) ≤ χ [0,T ] (t -τ) E(t,t -τ ; n) -U(t,t -τ) f (t -τ) .
Hence, this implies

sup τ≥0 (e -τK 0 /n e -τB/n ) n f -e -τK f 2 ≤ sup τ≥0 T 0 χ [0,T ] (t -τ) E(t,t -τ ; n) -U(t,t -τ) 2 f (t -τ) 2 dt ≤ T 0 sup (t,s)∈ ∆ E(t, s ; n) -U(t, s) 2 f (r) 2 dr.
Using (2.15) we immediately obtain (2.20). Similarly (2.21) follows from [START_REF] Ichinose | Error estimate in operator norm of exponential product formulas for propagators of parabolic evolution equations[END_REF]. ✷

Comments

Section 2.1

The operator-norm convergence rate O(1/n) of [START_REF] Ichinose | Note on the paper: "The norm convergence of the Trotter-Kato product formula with error bound[END_REF] for pairs of non-negative selfadjoint operators is sharp and ultimate optimal due to observations in [START_REF] Tamura | A remark on operator-norm convergence of Trotter-Kato product formula[END_REF]. The same remark concerns the sharpness and optimality of the rate O(1/n 2α-1 ) obtained first in [START_REF] Neidhardt | Fractional powers of self-adjoint operators and Trotter-Kato product formula[END_REF] under assumption (2.5) together with the A α -smallness of B α with relative bound less then one. Then the same ultimate sharp rate was proven in [START_REF] Ichinose | Trotter-Kato product formula and fractional powers of self-adjoint generators[END_REF], when the smallness condition is relaxed to the mild subordination (2.6). It is an open problem whether the assumption (2.6) is really necessary.

Section 2.2

It is unclear whether the convergence rates O(ln(n)/n) and O(ln(n)/n α ) are sharp. One expects convergence rates identical to that in Section 2.1.

Section 2.3

The approach used here was developed in [START_REF] Neidhardt | Convergence rate estimates for Trotter product approximations of solution operators for non-autonomous Cauchy problems[END_REF][START_REF] Neidhardt | On abstract linear evolution equations[END_REF][START_REF] Neidhardt | On abstract linear evolution equations[END_REF][START_REF] Neidhardt | On linear evolution equations. III: Hyperbolic case[END_REF][START_REF] Neidhardt | Linear non-autonomous Cauchy problems and evolution semigroups[END_REF][START_REF] Nickel | On evolution semigroups and nonautonomous Cauchy problems[END_REF]. The idea is to transform a time-dependent evolution problem to a time-independent problem, see also the next section.

Let us add some remarks. One easily checks that the operator K 0 is not selfadjoint whereas the operator B is self-adjoint. However, K 0 is maximal accretive. This is in some sense in contrast to Section 2.2, where the pair {A, B} consists of a self-adjoint operator A and an maximal accretive operator B such that dom(A) ⊆ dom(B) and dom(A) ⊆ dom(B * ). In the evolution case the conditions dom(K 0 ) ⊆ dom(B) and dom(K * 0 ) ⊆ dom(B) are satisfied but in the reversed order with respect to Section 2.2. However, the convergence rate O(ln(n)/n) is not affected by this.

The proof of the estimate (2.14) and (2.15) are very involved. Naturally the problem arises whether one can give a direct proof the estimate (2.14) avoiding those propagator estimates. [START_REF] Cachia | Operator-norm approximation of semigroups by quasisectorial contractions[END_REF] Trotter product formula on Banach spaces

Holomorphic case

There are only few generalisations of the results of Section 2 to Banach spaces. The main obstacle for that is the fact that the concept of self-adjointness is missing in the Banach spaces. One of solution is to relax the self-adjointness replacing the non-negative self-adjoint generator A by a generator of the holomorphic semigroup. The following result was proved in [START_REF] Cachia | Operator-norm convergence of the Trotter product formula for holomorphic semigroups[END_REF]. (e -τB/n e -τA/n ) ne -τC = O((ln(n)) 2 /n), and sup

τ∈[0,T ] (e -τA/n e -τB/n ) n -e -τC = O((ln(n)) 2 /n),
for any T > 0.

Evolution case

Similarly of Section 2.3 let us consider a generator A of a holomorphic semigroup on the separable Banach space X and a family of {B(t)} t∈[0,T ] of generators of holomorphic semigroups on X. We make the following assumptions:

Assumption 3.3 (A1)
The operator A is a generator of a holomorphic contraction semigroup on X such that 0 ∈ ρ(A). (A2) Let {B(t)} t∈[0,T ] be a family of closed operators such that for a.e. t ∈ [0, T ] and some α ∈ (0, 1) the condition dom(A α ) ⊂ dom(B(t)) is satisfied such that

C α := ess sup t∈[0,T ] B(t)A -α B(X) < ∞ .
(A3) Let {B(t)} t∈[0,T ] be a family of generators of contraction semigroups in X such that the function [0, T ] ∋ t → (B(t) + ξ ) -1 x ∈ X is strongly measurable for any x ∈ X and any ξ > b > 0.

(A4) We assume that dom(A * ) ⊂ dom(B(t) * ) and

C * 1 := ess sup t∈[0,T ] B(t) * (A * ) -1 B(X * ) < ∞,
where A * and B(t) * denote operators which are adjoint of A and B(t), respectively. (A5) There exists β ∈ (α, 1) and a constant L β > 0 such that for a.e. t, s ∈ [0, T ] one has the estimate:

A -1 (B(t) -B(s))A -α ≤ L β |t -s| β .
(A6) There exists a constant L 1 > 0 such that for a.e. t, s ∈ [0, T ] one has the estimate:

A -α (B(t) -B(s))A -α ≤ L 1 |t -s| .
The assumption 0 ∈ ρ(A) in (A1) is made for simplicity. We note that assumption (A3) is similar to (2.7). Assumption (A4) is automatically satisfied for self-adjoint operators. Assumption (A5) is a modification of (2.13) while assumption (A6) coincides with (2.13).

With the family {C(t)} t∈[0,T ] , C(t) = A + B(t), one associates the evolution equation (2.8). It turns out that under the assumptions (A1) and (A2) the family {C(t)} t∈[0,T ] consists of generators of contraction semigroups.

In accordance with Section 2.3 we consider the Banach space L p ([0, T ], X) for some fixed p ∈ [0, 1) and introduce the multiplication operators A and B, cf. (2.17) and (2.18). Under the assumptions (A1)-(A3) the operators A and B are generators of contraction semigroups such that dom(A ) ⊆ dom(B), in particular, A is a generator of holomorphic semigroup. Similarly, one can introduce the multiplication operator C induced by the family {C(t)} t∈[0,T ] which is also a generator of a holomorphic semigroup. Notice that C = A + B and dom(C ) = dom(A ).

Let D 0 the generator of the right-shift semigroup on L p ([0, T ], X), i.e.

(e -τD 0 f

)(t) = f (t -τ)χ [0,T ] (t -τ) f (t -τ), f ∈ L p ([0, T ], X),
cf. (2.19). We consider the operator

K f = D 0 f + A f + B f , f ∈ dom( K ) = dom(D 0 ) ∩ dom(A ) ∩ dom(B),
cf. (2.19). Assuming (A1)-(A3) it was shown in [START_REF] Neidhardt | Convergence rate estimates for Trotter product approximations of solution operators for non-autonomous Cauchy problems[END_REF] that the operator K is closable and its closure K is the generator of a semigroup. Furthermore, we set

K 0 f = D 0 f + A f , f ∈ dom( K 0 ) = dom(D 0 ) ∩ dom(A ),
In contrast to the Hilbert space the operator K 0 is not necessary a generator of a semigroup. However, the operator K 0 closable and its closure K 0 is a generator. Notice that K coincides with the algebraic sum of K 0 and B, i.e K = K 0 + B.

In [START_REF] Neidhardt | Convergence rate estimates for Trotter product approximations of solution operators for non-autonomous Cauchy problems[END_REF] the following theorem was proved. when n → ∞.

Assuming instead of assumption (A5) the assumption (A6) the result slightly modifies, see [START_REF] Neidhardt | On convergence rate estimates for approximations of solution operators for linear non-autonomous evolution equations[END_REF].

Theorem 3.5 ([14, Theorem 5.4]) Let the assumptions (A1)-(A4) be satisfied for some α ∈ (1/2, 1). If (A6) is valid, then for n → ∞ one gets the asymptotic:

sup τ≥0 (e -τB/n e -τK 0 /n ) n -e -τK = O(1/n 1-α ).
(3.25)

Convergence rates for propagators

The proof of both theorems does not use propagator approximations of type (2.14) or (2.15). However, Theorem 3.4 and 3.5 can be used to prove propagator approximations. To this end one has to introduce the notion of a evolution semigroup.

Definition 3.6 A generator K in L p ([0, T ], X), p ∈ [1, ∞), is called a evolution generator if (i) dom(K ) ⊂ C([0, T ], X) and M(φ )dom(K ) ⊂ dom(K ) for φ ∈ W 1,∞ ([0, T ]), (ii) K M(φ ) f -M(φ )K f = M( φ ) f for f ∈ dom(K ) and φ ∈ W 1,∞ ([0, T ])
, where φ = ∂ t φ , (iii) the domain dom(K ) has a dense cross-section, i.e. for each t ∈ (0, T ] the set

[dom(K )] t := {x ∈ X : ∃ f ∈ dom(K ) such that x ∈ f (t)}, is dense in X. By M(φ ), L ∞ ([0, T ]), the bounded operator (M(φ ) f )(t) = φ (t) f (t), f ∈ L p ([0, T ], X). is meant.
One can check that the operator K defined as the closure of K is an evolution generator, cf. [START_REF] Neidhardt | Convergence rate estimates for Trotter product approximations of solution operators for non-autonomous Cauchy problems[END_REF]Theorem 1.2]. Evolution generators a directly related to propagators. For this purpose one has slightly weaken the notion of a propagator defined in Section 2.3. Definition 3.7 Let {U(t, s)} (t,s)∈∆ , ∆ = {(t, s) ∈ (0, T ] × (0, T ] : s ≤ t ≤ T }, be a strongly continuous family of bounded operators on X. If the conditions U(t,t) = I for t ∈ (0, T ] ,

(3.26)

U(t, r)U(r, s) = U(t, s) for t, r, s ∈ (0, T ] with s ≤ r ≤ t , (3.27) U B(X) := sup (t,s)∈∆ U(t, s) < ∞ (3.28)
are satisfied, then {U(t, s)} (t,s)∈∆ is called a propagator.

Comparing with Section 2.3 we note that ∆ slightly differs from ∆ . Indeed, ∆ ⊆ ∆ but ∆ = ∆ . Restricting (2.9) and (2.10) to (0, T ] we get (3.26) and (3.27), respectively. Condition (3.28) is necessary because the set ∆ is not closed.

It is known that there is an one-to-one correspondence between the set of all evolution generators on L p ([0, T ], X) and the set of all propagators in the sense of Definition 3.7 established by

(e -τK f )(t) = U(t,t -τ)χ [0,T ] (t -τ) f (t -τ), f ∈ L p ([0, T ], X),
cf. [START_REF] Neidhardt | Convergence rate estimates for Trotter product approximations of solution operators for non-autonomous Cauchy problems[END_REF]Theorem 3.3] or [START_REF] Neidhardt | On abstract linear evolution equations[END_REF]Theorem 4.12].

Let K 0 be the generator of an evolution semigroup {U 0 (τ)} τ≥0 and let B be a multiplication operator induced by a measurable family {B(t)} t∈[0,T ] of generators of contraction semigroups. Note that in this case the multiplication operator B is a generator of a contraction semigroup (e -τ B f )(t) = e -τ B(t) f (t), on the Banach space L p ([0, T ], X). Since {U 0 (τ)} τ≥0 is an evolution semigroup, then there is a propagator {U 0 (t, s)} (t,s)∈∆ such that the representation

(U 0 (τ) f )(t) = U 0 (t,t -τ)χ [0,T ] (t -τ) f (t -τ), f ∈ L p ([0, T ], X),
is valid for a.e. t ∈ [0, T ] and τ ≥ 0. Then we define

Q j (t, s; n) := U 0 (s + j (t-s) n , s + ( j -1) (t-s) n )e -(t-s) n B s+( j-1) (t-s) n
where j ∈ {1, 2, . . . , n}, n ∈ N, (t, s) ∈ ∆ , and we set

V n (t, s) := n ← ∏ j=1 Q j (t, s; n), n ∈ N, (t, s) ∈ ∆ ,
where the product is increasingly ordered in j from the right to the left. Then a straightforward computation shows that the representation

e -τK 0 /n e -τB/n n f (t) = V n (t,t -τ)χ [0,T ] (t -τ) f (t -τ) , f ∈ L p ([0, T ], X), holds for each τ ≥ 0 and a.e. t ∈ [0, T ]. Similarly we can introduce G j (t, s; n) = e -t-s n B(s+ j t-s n ) U 0 (s + j t-s n , s + ( j -1) t-s n ) where j ∈ {1, 2, . . . , n}, n ∈ N, (t, s) ∈ ∆ . Let U n (t, s) := n ← ∏ j=1 G j (t, s; n), n ∈ N, (t, s) ∈ ∆ ,
where the product is again increasingly ordered in j from the right to the left. We verify that Theorems 3.4 and 3.5 are proved in [START_REF] Neidhardt | Convergence rate estimates for Trotter product approximations of solution operators for non-autonomous Cauchy problems[END_REF] and [START_REF] Neidhardt | On convergence rate estimates for approximations of solution operators for linear non-autonomous evolution equations[END_REF] under the assumption that the pair {K 0 , B} is Trotter-stable. The convergence rates (3.24) and (3.25) differ significantly from the convergence rate O(ln(n)/n) of Proposition 2.1. It is an open problem whether the convergences rates (3.24) and (3.25) can be improved to O(ln(n)/n). One has to mention that the convergence (3.25) coincides with that one of (3.23) despite the fact that K 0 is not a generator of a holomorphic semigroup. Indeed, the generator K 0 is not holomorphic since e -τK 0 = 0 for τ ≥ T .

e -τB/n e -τK 0 /n n f (t) = U n (t,t -τ)χ [0,T ] (t -τ) f (t -τ) , f ∈ L p ([0, T ], X),

Section 3.3

It is a bit surprising that the operator-norm convergence of the Trotter product formula for the pairs: generator of evolution semigroup and multiplication operator, is equivalent to the operator-norm convergence of a certain approximation of the corresponding propagator, see Proposition 3.8. In particular, this yields that two convergences: (2.15) and (2.20), are equivalent.

Sharpness

Example

Let us consider a "solvable" example. We study bounded perturbations of the evolution generator D 0 . To do this aim we consider X = C and we denote by ]). Then, q induces a bounded multiplication operator Q in the Banach space L 2 ([0, 1]):

L 2 ([0, 1]) the Hilbert space L 2 ([0, T ], C). For t ∈ [0, 1], let q : t → q(t) ∈ L ∞ ([0, 1
(Q f )(t) = q(t) f (t), f ∈ L 2 ([0, 1]).
For simplicity we assume that q ≥ 0. Then Q generates on L p ([0, 1]) a contraction semigroup {e -τQ } τ≥0 . Since generator Q is bounded, the closed operator K := D 0 + Q, with domain dom(K ) = dom(D 0 ), is generator of a semigroup on L p ([0, 1]). From [START_REF] Trotter | On the product of semi-groups of operators[END_REF] one gets immediately s-lim n→∞ e -τD 0 /n e -τQ/n n = e -τ(D 0 +Q) uniformly in τ ∈ [0, T ] for any T > 0. One easily checks that K is an evolution generator. A straightforward computation shows that e -τ(D 0 +Q) f (t) = e -t t-τ q(y)dy χ [0,1] (t -τ) f (t -τ)

which yields that the propagator corresponding to K is given by U(t, s) = e -t s q(y)dy , (t, s) ∈ ∆ .

A simple computation shows that

e -τD 0 /n e -τQ/n n f (t) =: V n (t,t -τ)χ [0,T ] (t -τ) f (t -τ) .
Then by straightforward calculations one finds that

V n (t, s) = e -t-s n ∑ n-1 k=0 q(s+k t-s n ) , (t, s) ∈ ∆ . Proposition 4.1 ([13, Proposition 3.1]) Let q ∈ L ∞ ([0, T ]) be non-negative. Then sup τ≥0 e -τ(D 0 +Q) -e -τD 0 /n e -τQ/n n B(L p ([0,1])) = Θ ess sup (t,s)∈∆ t s q(y)dy - t -s n n-1 ∑ k=0 q(s + k t-s n )
as n → ∞, where Θ is the Landau symbol defined in the introduction.

Note that by Proposition 3.8 the operator-norm convergence rate of the Trotter product formula for the pair {D 0 , Q} coincides with the convergence rate of the integral Darboux-Riemann sum approximation of the Lebesgue integral.

Results

Below we give a series of examples which show the dependence of the convergence rate on the smoothness of the function q ∈ L ∞ ([0, T ]). First we consider the Hölder and Lipschitz continuous cases.

∈ C 0,β ([0, T ]), β ∈ (0, 1], is non-negative, then for n → ∞ one gets sup τ≥0 e -τ(D 0 +Q) -e -τD 0 /n e -τQ/n n = O(1/n β ) .
Now a natural question that one may to ask is: what happens, when q is simply continuous? We comment that for a general continuous q one can say nothing about the Trotter product formula convergence rate. Indeed, as it follows from the next theorem the convergence to zero in (4.30) may be arbitrary slow. = ω(δ n ) , as n → ∞, where ω is the Landau symbol defined in the Introduction.

Our final comment concerns the case when q is only measurable. Then it can happen that the Trotter product formula for that pair {D 0 , Q} does not converge in the operator-norm topology: We note that Theorem 4.5 does not exclude the convergence of the Trotter product formula for the pair {D 0 , Q} in the strong operator topology.

Comments

Section 4.1

Our example can be considered as a kind of solvable model. It fits into the evolution cases considered in Sections 2.3 and 3.2. Indeed, one has to set X = C, p = 2, L p ([0, T ], X) = L 2 ([0, T ]), A = I and B(t) = q(t) and B = Q. One easily checks that (e -τ/nK 0 e -τB/n ) n = e -τ (e -τ/nD 0 e -τQ/n ) n which shows that the convergence rate of (e -τ/nD 0 e -τQ/n ) n coincides with that one (e -τ/nK 0 e -τB/n ) n . One easily checks that the choice A = I, B(t) = q(t), q(t) ≥ 0, guarantees the assumptions (A1)-(A4) If q ∈ C 0,β ([0, T ]), then the assumption (A5) is satisfied. If Q is Lipschitz continuous, i.e. q ∈ C 0,1 ([0, T ]), then the assumption (A6) is valid. Section 4.2 Theorem 4.2 shows that for β ∈ (0, 1) the convergence rate is O(1/n β ), which is better than the convergence rates O(1/n β -α ) and O(1/n 1-α ) in Theorems 3.4 and 3.5. Hence, they are not sharp. Theorems 4.3 and 4.4 demonstrate that the convergence rate can be arbitrary slow if the smoothness of Q is weaker and weaker.

Finally, Theorem 4.5 shows that there is a bounded operator such that the Trotter product formula does not converge in the operator norm. This makes clear that Theorem 3.2 becomes false if the condition that A is a holomorphic generator is dropped. Indeed the operator D 0 which plays the role of A of Theorem 3.2 is not a generator of holomorphic semigroup.

Theorem 3 . 1 ([ 5 ,

 315 Theorem 3.6 and Corollary 3.7]) Let A be a generator of a holomorphic contraction semigroup on the separable Banach space X and let B a generator of a contraction semigroup on X. (i) If for some α ∈ (0, 1) the conditiondom(A α ) ⊆ dom(B) ,holds and dom(A * ) ⊆ dom(B * ) is satisfied, then the operator sum C = A + B is a generator of a contraction semigroup and sup τ∈[0,T ] (e -τB/n e -τA/n ) ne -τC = O(ln(n)/n 1-α ) , (3.22) for any T > 0. (ii) If for some α ∈ (0, 1) the condition dom((A α ) * ) ⊆ dom(B * ) , is satisfied and dom(A) ⊆ dom(B) is valid, then C = A + B is the generator of a contraction semigroup andsup τ∈[0,T ] (e -τA/n e -τB/n ) ne -τC = O(ln(n)/n 1-α ) ,(3.23)for any T > 0. Theorem 3.2 ([5, Theorem 3.6 and Corollary 3.7]) Let A be a generator of a holomorphic contraction semigroup on X and let B a generator of a contraction semigroup on X. If B in addition a bounded operator, then sup τ∈[0,T ]

Theorem 3 . 4 ([ 12 ,

 3412 Theorem 7.8]) Let the assumptions (A1)-(A4) be satisfied for some α ∈ (0, 1). If (A5) holds, then sup τ≥0 (e -τB/n e -τK 0 /n ) ne -τK = O(1/n β -α ),(3.24)

Theorem 4 . 2 (

 42 [START_REF] Neidhardt | Remarks on the operator-norm convergence of the Trotter product formula[END_REF] Theorem 3.2]) If the function: q

Theorem 4 . 3 (

 43 [START_REF] Neidhardt | Remarks on the operator-norm convergence of the Trotter product formula[END_REF] Theorem 3.3]) If q : [0, 1] → C, is continuous and non-negative, then for n → ∞ e -τ(D 0 +Q)e -τD 0 /n e -τQ/n n = o(1) . (4.30)

Theorem 4 . 4 (

 44 [START_REF] Neidhardt | Remarks on the operator-norm convergence of the Trotter product formula[END_REF] Theorem 3.4]) Let δ n > 0 be a sequence with δ n → 0 as n → ∞. Then there exists a continuous function q : [0, 1] → R such that sup τ≥0 e -τ(D 0 +Q)e -τD 0 /n e -τQ/n n B(L p ([0,1]))

Theorem 4 . 5 (

 45 [START_REF] Neidhardt | Remarks on the operator-norm convergence of the Trotter product formula[END_REF] Theorem 3.5]) There is a non-negative measurable function q ∈ L ∞ ([0, 1]), such that lim inf n→∞ sup τ≥0 e -τ(D 0 +Q)e -τD 0 /n e -τQ/n n B(L p ([0,1])) > 0 .

  Proposition 2.1 Let A be a non-negative self-adjoint operator on the separable Hilbert space H and let {B(t)} t∈[0,T ] be a family of non-negative self-adjoint operators. If the assumptions (2.7) and (2.13) are satisfied, then

	sup
	τ≥0

  holds for each τ ≥ 0 and a.e. t ∈ [0, T ]. Proposition 3.8 ([13, Proposition 2.1]) Let K and K 0 be generators of evolution semigroups on the Banach space L p ([0, T ], X) for some p ∈ [1, ∞). Further, let {B(t))} t∈[0,T ] be a strongly measurable family of generators of contraction on X. ∈ ∆ , with increasingly ordered product in j from the right to the left. From Theorem 3.4 and Proposition 3.8 one immediately obtains the following theorem. It is unclear whether Theorem 3.1 is sharp. Theorem 3.1 should be valid if the contractivity of the involved semigroups is replacedTrotter-stability. A pair of generators {A, B} is called Trotter-stable if the Trotter product is uniformly bounded in n ∈ N, i.e., if sup It turns out that the pair {A, B} is Trotter-stable if and only {B, A} is Trotter-stable.

	3.4 Comments
	Section 3.1
				sup	e -τA/n e -τB/n n	< ∞.
		n∈N	τ≥0
	Section 3.2
	Then
	sup τ≥0	e -τK -e -τK 0 /n e -τB/n n	= ess sup (t,s)∈∆	U(t, s) -V n (t, s) , n ∈ N,
	and
	sup τ≥0	e -τK -e -τB/n e -τK 0 /n n	= ess sup (t,s)∈∆	U(t, s) -U n (t, s) , n ∈ N.
	Let us introduce the approximations
			n←
		U n (t, s) :=	j=1 ∏	G j (t, s ; n), n = 1, 2, . . . ,
		G j (t, s ; n) := e -t-s n B(s+ j t-s n ) e -t-s n A , j = 0, 1, 2, . . . , n,
	(t, s) Theorem 3.9 ([12, Theorem 1.4]) Let the assumptions (A1)-(A4) be satisfied. If
	(A5) holds, then for n → ∞ the rate:
		ess sup (t,s)∈∆		U n (t, s) -U(t, s) = O(1/n β -α ) .	(3.29)
	From Theorem 3.5 and Proposition 3.8 we get
	Theorem 3.10 ([14, Theorem 5.6]) Let the assumptions (A1)-(A4) be satisfied for
	some α ∈ (1/2, 1). If (A6) is valid, then for n → ∞ one obtains a better rate:
		ess sup (t,s)∈∆		U n (t, s) -U(t, s) = O(1/n 1-α ) .