Alexander Borichev

AND EL A N Le

Hassan Youssfi
email: el-hassan.youssfi@univ-amu.fr

El Hassan

ON THE DIMENSION OF THE FOCK TYPE SPACES

Keywords: Fock space, subharmonic function, plurisubharmonic function

We study the weighted Fock spaces in one and several complex variables. We evaluate the dimension of these spaces in terms of the weight function extending and completing earlier results by Rozenblum-Shirokov and Shigekawa. C dµ(z) = ∞,

Introduction

Let ψ be a plurisubharmonic function on C n , n ≥ 1. The weighted Fock space F 2 ψ is the space of entire functions f such that

f 2 ψ = C n |f (z)| 2 e -ψ(z) dv(z) < ∞,
where dv is the volume measure on C n . Note that F 2 ψ is a closed subspace of L 2 (C n , e -ψ dv) and hence is a Hilbert space endowed with the inner product f, g ψ = C n f (z)g(z)e -ψ(z) dv(z), f, g ∈ F 2 ψ .

In this paper we study when the space F 2 ψ is of finite dimension depending on the weight ψ. This problem (at least for the case n = 1) is motivated by some quantum mechanics questions, especially by the study of zero modes, eigenfunctions with zero eigenvalues.

In [START_REF] Rozenblum | Infiniteness of zero modes for the Pauli operator with singular magnetic field[END_REF]Theorem 3.2], Rozenblum and Shirokov proposed a sufficient condition for the space F 2 ψ to be of infinite dimension, when ψ is a subharmonic function.

More precisely, they claimed that if ψ is a finite subharmonic function on the complex plane such that the measure µ = ∆ψ is of infinite mass:

(1. [START_REF] Bedford | The Dirichlet problem for a complex Monge-Ampère equation[END_REF] µ(C) = then the space F 2 ψ has infinite dimension. (For the fact that if µ = ∆ψ a non-trivial doubling measure, then F 2 ψ has infinite dimension see [START_REF] Haslinger | Complex analysis. A functional analytic approach[END_REF]Theorem 11.45]). We improve and extend somewhat the statement of Rozenblum-Shirokov in our paper, give a necessary and sufficient condition on ψ for the space F 2 ψ to be of finite dimension, and calculate this dimension. The situation is much more complicated in C n , n ≥ 2. Shigekawa established in [START_REF] Shigekawa | Spectral properties of Schrödinger operators with magnetic fields for a spin 1/2 particle[END_REF] (see also [START_REF] Haslinger | Complex analysis. A functional analytic approach[END_REF]Theorem 11.20] in a book by Haslinger), the following interesting result.

Theorem A. Let ψ : C n → R be a C ∞ smooth function and let λ 0 (z) be the smallest eigenvalue of the Levi matrix

L ψ (z) = i∂ ∂ψ(z) = ∂ 2 ψ(z) ∂z j ∂z k n j,k=1
.

Suppose that

(1.2) lim |z|→∞ |z| 2 λ 0 (z) = ∞.
Then dim F 2 ψ = ∞. Note that the condition (1.2) is not necessary. A corresponding example is given in [4, Section 11.5] (ψ(z, w) = |z| 2 |w| 2 + |w| 4 ). In this paper, we improve Theorem A by presenting a weaker condition for the dimension of the Fock space F 2 ψ to be infinite. Furthermore, we give several examples that show how far is our condition from being necessary. Finally, we consider several examples (classes of examples) of weight functions ψ of special form and evaluate the dimension of F 2 ψ . The rest of the paper is organised as follows. The case of dimension one is considered in Section 2, and the case of higher dimension is considered in Section 3.

The case of C

Given a subharmonic function ψ : C → [-∞, ∞) denote by µ ψ the corresponding Riesz measure, µ ψ = ∆ψ. Next, consider the class M d of the positive σ-finite atomic measures with masses which are integer multiples of 4π. Given a σ-finite measure µ, consider the corresponding atomic measure µ d ,

µ d = max µ 1 ∈ M d : µ 1 ≤ µ .
In fact, for every atom aδ x of µ, µ d has at the point x an atom of size 4π times the integer part of a/(4π). Denote

µ c = µ -µ d , µ d = k 4πδ x k,µ .
Denote by M c the class of the positive σ-finite measures µ such that µ d = 0. Note that if ψ is finite on the complex plane, then µ ψ has no point masses and

µ ψ ∈ M c . Furthermore, if µ ψ ∈ M c , then e -ψ ∈ L 1 loc (v). Lemma 2.1. Let ψ, ψ 1 be two subharmonic functions such that (µ ψ ) c = (µ ψ 1 ) c . Then dim F 2 ψ = dim F 2 ψ 1 . Proof.
Let F, F 1 be two entire functions with the zero sets, correspondingly, {x k,µ ψ } and {x k,µ ψ 1 } (taking into account the multiplicities). Then ∆ log

|F | 2 = (µ ψ ) d , ∆ log |F 1 | 2 = (µ ψ 1 ) d ,

and the functions

h = ψ -log |F 2 | -ψ c , h 1 = ψ 1 -log |F 2 1 | -ψ c 1 are harmonic. Let h = ℜH, h 1 = ℜH 1 for some entire functions H, H 1 .
Given an entire function f we have

f ∈ F 2 ψ ⇐⇒ C |f (z)| 2 e -ψ(z) dv(z) < ∞ ⇐⇒ C |f (z)| 2 e -ψ c (z)-h(z)-log |F (z)| 2 dv(z) < ∞ ⇐⇒ C |f (z)e -H(z)/2 /F (z)| 2 e -ψ c (z) dv(z) < ∞ ⇐⇒ C |f (z)e -H(z)/2 /F (z)| 2 e -ψ c 1 (z) dv(z) < ∞ ⇐⇒ C |f (z)e -H(z)/2 /F (z)| 2 e -ψ 1 (z)+h 1 (z)+log |F 1 (z)| 2 dv(z) < ∞ ⇐⇒ C |f (z)e -H(z)/2+H 1 (z)/2 |F 1 (z)/F (z)| 2 e -ψ 1 (z) dv(z) < ∞ ⇐⇒ f • F 1 F e -H/2+H 1 /2 ∈ F 2 ψ 1 . Thus, dim F 2 ψ = dim F 2 ψ 1 . Lemma 2.2. Let ψ be a subharmonic function such that µ ψ ∈ M c . If dim F 2 ψ < ∞, then µ ψ (C) < ∞. See the proof of [8, Theorem 3.2]. Lemma 2.3. Let ψ be a subharmonic function. Then dim F 2 ψ ≤ µ ψ (C) 4π .
Here and later on, given a real number x, ⌈x⌉ is the maximal integer smaller than x.

Proof. Set µ = µ ψ and consider a modified logarithmic potential G of the measure µ:

G(z) = 1 2π D(0,2) log|z -w| dµ(w) + 1 2π C\D(0,2) log z -w w dµ(w) = G 1 (z) + G 2 (z).
Here and later on, D(z, r) = {w ∈ C : |w -z| < r}. Since ∆G = µ = ∆ψ, by Lemma 2.1 we have dim

F 2 ψ = dim F 2 G . Next, (2.1) G 1 (z) - µ(D(0, 2)) 2π log |z| ≤ 1 2π D(0,2) log 1 - w z dµ(w) ≤ C |z| , |z| ≥ 4,
and

G 2 (z) - µ(C \ D(0, 2)) 2π log |z| = 1 2π C\D(0,2) log 1 z - 1 w dµ(w) ≤ 0, |z| ≥ 4.
Thus,

G(z) ≤ µ(C) 2π log(1 + |z|) + C 1 + |z| , z ∈ C.
Now, given an entire function f , we have

f ∈ F 2 ψ =⇒ C |f (z)| 2 (1 + |z|) -µ(C)/(2π) dv(z) < ∞.
By a Liouville type theorem, f is a polynomial of degree N such that

∞ 1 r 2N r -µ(C)/(2π) rdr < ∞. Therefore, N < -1 + µ(C)/(4π). Thus, dim F 2 ψ ≤ µ(C) 4π .
Lemma 2.4. Let ψ be a subharmonic function and suppose that µ ψ ∈ M c . Then

dim F 2 ψ ≥ µ ψ (C) 4π . Proof. Set µ = µ ψ and choose ε > 0, R > 1 such that µ(D(0, R)) 4π > µ(C) 4π + ε 2 .
Next, increasing R, we can guarantee that

µ(D(0, R)) > µ(C) - 1 2
.

Consider a modified logarithmic potential U of measure µ:

U(z) = 1 2π D(0,R) log|z -w| dµ(w) + 1 2π C\D(0,R) log z -w w dµ(w) = U 1 (z) + U 2 (z).
Since ∆U = µ = ∆ψ, by Lemma 2.1 we have dim

F 2 ψ = dim F 2 U .
Arguing as in (2.1), we get

U 1 (z) ≥ µ(D(0, R)) 2π log |z| - C |z| , |z| ≥ 2R. Next, let |z| ≥ 2R. Then U 2 (z) = 1 2π C\(D(0,R)∪D(z,|z|/2)) log z -w w dµ(w) + 1 2π D(z,|z|/2) log z -w w dµ(w) ≥ C - 1 2π D(z,|z|/2) log z/2 z -w dµ(w) = C -U 3 (z).
Now, we apply a result by Hayman [5, Lemma 4]. The following notation is used there. Let ν be a finite positive measure. Given

z ∈ C, h > 0, set n(z, h) = ν(D(z, h)), N(z, h) = D(z,h) log h w-z dν(w). Lemma 2.5. Let z 0 ∈ C, 0 < d < h/2.
There exists a set S of area at most πd 2 such that

N(z, h/2) ≤ n(z 0 , h) log 16h d , z ∈ D(z 0 , h/2) \ S. Given m ≥ 1, denote A m = {z ∈ C : 2 m R ≤ |z| < 2 m+1 R}. Fix m ≥ 1 and k ≥ 1 and apply Lemma 2.5 with ν = 1 C\D(0,R) µ, 2 m R ≤ |z 0 | < 2 m+1 R, h = 2 m-1 R, n(z 0 , h) ≤ 1/2, and d = 2 m-k-1 R to get for some C, C 1 > 0, δ ∈ (0, 1): m 2 z ∈ A m : U 3 (z) > C 1 + δk ≤ C • 2 2m R 2 2 -2k , k ≥ 1. Hence, C (1 + |z|) -2-ε e U 3 (z) dv(z) ≤ C + C m≥1 k≥1 2 -(2+ε)m e δk × m 2 z ∈ A m : C 1 + δk ≤ U 3 (z) < C 1 + δ(k + 1) ≤ C + C m≥1 k≥1 2 -(2+ε)m e δk 2 2m R 2 2 -2k < ∞. Next, for every 0 ≤ N ≤ µ(C) 4π -1 we have C |z| 2N e -U (z) dv(z) ≤ C C |z| 2N (1 + |z|) -µ(D(0,R))/(2π) e U 3 (z) dv(z) ≤ C C (1 + |z|) -2-ε e U 3 (z) dv(z) < ∞
Here we use that µ ψ ∈ M c and, hence, e -U is locally integrable. Finally, we have

dim F 2 ψ ≥ µ(C) 4π .
Summing up Lemmata 2.1, 2.2, 2.3, and 2.4, we obtain the following result, extending and slightly correcting [8, Theorem 3.2].

Theorem 2.6. Let ψ be a subharmonic function on the complex plane. Then the Fock space F 2 ψ is finite-dimensional if and only if

(2.2) (µ ψ ) c (C) < ∞.
If ψ is finite on C, then we can write condition (2.2)

as µ ψ (C) < ∞. Finally, if (µ ψ ) c (C) < ∞, then dim F 2 ψ = (µ ψ ) c (C) 4π .
Remark 2.7. It is an interesting open question to characterize non subharmonic functions ψ such that the space F 2 ψ is of finite dimension. For some results in this direction and some physical interpretations see [START_REF] Rozenblum | Entire functions in weighted L 2 and zero modes of the Pauli operator with non-sign definite magnetic field[END_REF].

The case of C

n , n > 1 Let C n denote the n-dimensional complex Euclidean space. Given z = (z 1 , z 2 , . . . , z n ) ∈ C n , we set |z| = |z 1 | 2 + • • • + |z n | 2 . Denote B n (z, r) = {w ∈ C n : |w -z| < r}. Then B n = B n (0, 1)
is the unit ball and S n = ∂B n is the unit sphere in C n . Let dσ be the normalized surface measure on S n .

Theorem 3.1. Let ψ : C n → R be a C 2 smooth function. Given M > 0, consider ψ M (z) = M log(|z| 2 ). Suppose that for every M > 0, the function ψ -ψ M is plurisubharmonic outside a compact subset of C n . Then dim F 2 ψ = ∞.
Proof. We use the fundamental result of Bedford-Taylor [START_REF] Bedford | The Dirichlet problem for a complex Monge-Ampère equation[END_REF] on the solutions of the Dirichlet problem for the complex Monge-Ampère equation. Given M > 0, choose r M > 1 such that ψ -ψ M is plurisubharmonic on C n \B n (0, r M ). Solving the Dirichlet problem for the complex Monge-Ampère equation on B n (0, r M ) with the boundary conditions (ψ -ψ M )| ∂Bn(0,r M ) , we obtain a function u. Set

ψ M (z) = (ψ -ψ M )(z), z ∈ C n \ B n (0, r M ), u(z), z ∈ B n (0, r M ).
Then ψ M is a continuous plurisubharmonic function on C n (see also [START_REF] Demailly | Potential Theory in Several Complex Variables[END_REF]Section 7]). Now, by the Hörmander theorem ([6, Theorem 4.4.4], see also [2, Section IV]), there exists an entire function f ≡ 0 such that

C n |f (z)| 2 (1 + |z| 2 ) -3n e -ψ M (z) dv(z) < ∞.
Hence, for every 0 ≤ k ≤ M - 3 2 n, we have

C n |f (z)| 2 |z| 2k e -ψ(z) dv(z) ≤ C + C n \Bn(0,r M ) |f (z)| 2 |z| 2k e -ψ(z) dv(z) = C + C n \Bn(0,r M ) |f (z)| 2 |z| 2k e -ψ M (z) e -(ψ(z)-ψ M (z)) dv(z) ≤ C + C n \Bn(0,r M ) |f (z)| 2 |z| -3n e -ψ M (z) dv(z) < ∞.
Since M is arbitrary, we have dim

F 2 ψ = ∞.
Remark 3.2. Theorem A is an immediate corollary of Theorem 3.1. Indeed, an easy computation shows that if ψ(z) = ϕ(|z| 2 ), ϕ ∈ C 2 ((0, +∞)), then

∂ 2 ψ ∂z j ∂ zk (z) = ϕ ′′ (|z| 2 ) zj z k + ϕ ′ (|z| 2 )δ jk ,
where δ jk is the Kronecker delta symbol. This implies that

i∂ ∂ψ(z) = ϕ ′ (|z| 2 )I + ϕ ′′ (|z| 2 )z * z,
where

z * =   z1 . . . zn   , z * z = zj z k n j,k=1
. Note also that the spectrum of the matrix i∂ ∂ψ(z) is

(3.1) σ(i∂ ∂ψ(z)) = ϕ ′ (|z| 2 ), ϕ ′ (|z| 2 ) + |z| 2 ϕ ′′ (|z| 2 ) .
The first eigenvalue has multiplicity n -1 and the second one has multiplicity 1. Furthermore,

L ψ (z) = i∂ ∂ψ(z) = i∂ ∂(ψ -ψ M )(z) + M |z| 2 I - M |z| 4 z * z = L ψ-ψ M (z) + M |z| 2 I - M |z| 4 z * z. Let z ∈ C n and let V =   V 1 . . . V n 
 be a normalized eigenvector corresponding to an eigenvalue ν of L ψ-ψ M (z). By the hypothesis of Theorem A, for |z| > r M we have λ 0 (z)|z| 2 ≥ M, where λ 0 (z) is the smallest eigenvalue of L ψ (z). Thus,

ν = L ψ-ψ M (z)V, V = L ψ (z)V, V - M |z| 2 + M |z| 4 z * zV, V ≥ λ 0 (z) - M |z| 2 + M |z| 4 |zV | 2 ≥ 0.
Therefore, ψ -ψ M is plurisubharmonic on C n \ B n (0, r M ), and we are in the conditions of Theorem 3.1. Now we give an easy example when Theorem 3.1 applies while Theorem A does not work.

Example 3.3. Set ψ(z) = ϕ(|z| 2 ) = log(1 + |z| 2 ) 3/2 , z ∈ C n . Then ϕ(t) = log(1 + t) 3/2 , t > 0.
Evidently, dim F 2 ψ = ∞. We will show that condition (1.2) fails for ψ while the conditions of Theorem 3.1 are satisfied.

We have ϕ ′ (t) = 3 2

1 1 + t log(1 + t) 1/2 ,
and

ϕ ′′ (t) = - 3 2 log(1 + t) 1/2 (1 + t) 2 + 3 4(1 + t) 2 log(1 + t) 1/2 .
By (3.1), the eigenvalues of the matrix L ψ (z) are

λ 1 (z) = 3 log(1 + |z| 2 ) 1/2 2(1 + |z| 2 ) , and
λ 2 (z) = 3 log(1 + |z| 2 ) 1/2 2(1 + |z| 2 ) 2 + 3|z| 2 4(1 + |z| 2 ) 2 log(1 + |z| 2 ) 1/2 = 3 4 2 log(1 + |z| 2 ) + |z| 2 (1 + |z| 2 ) 2 log(1 + |z| 2 ) 1/2 .
For |z| ≥ 2, the smallest eigenvalue of the matrix L ψ (z) is λ 2 (z) and lim |z|→∞ |z| 2 λ 2 (z) = 0.

Thus, condition (1.2) does not hold. On the other hand, for M > 0, the eigenvalues of matrix

L ψ-ψ M (z) are α 1 (z) = λ 1 (z) - M |z| 2 , and α 2 (z) = λ 2 (z). Since lim |z|→∞ |z| 2 λ 1 (z) = ∞ and α 2 (z) > 0, z = 0, the conditions of Theorem 3.1 are satisfied.
In the rest of the paper we show that in different situations the sufficient condition of Theorem 3.1 is not necessary for dim

F 2 ψ = ∞. Example 3.4. Set ψ(z, w) = |z| 2 + 2 log(1 + |w| 2 ), w, z ∈ C.
It is clear that dim F 2 ψ = ∞. Let us verify that for M > 2 the function ψ -ψ M is not plurisubharmonic at the points (1, w), w ∈ C.

We start with some easy computations:

∂ψ ∂z = z, ∂ 2 ψ ∂z∂z = 1, ∂ 2 ψ ∂z∂w = 0, ∂ψ ∂w = 2w 1 + |w| 2 , ∂ 2 ψ ∂w∂z = 0, ∂ 2 ψ ∂w∂w = 2 (1 + |w| 2 ) 2 .
Now, given M > 0, we have

L ψ-ψ M (z, w) = 1 0 0 2 (1+|w| 2 ) 2 + M (|z| 2 + |w| 2 ) 2 |z| 2 zw zw |w| 2 - M |z| 2 + |w| 2 I = 1 -M |w| 2 (|z| 2 +|w| 2 ) 2 M zw (|z| 2 +|w| 2 ) 2 M zw (|z| 2 +|w| 2 ) 2 2 (1+|w| 2 ) 2 -M |z| 2 (|z| 2 +|w| 2 ) 2
, and, hence,

det(L ψ-ψ M (z, w)) = 2 (1 + |w| 2 ) 2 - M|z| 2 (|z| 2 + |w| 2 ) 2 - 2M|w| 2 (1 + |w| 2 ) 2 (|z| 2 + |w| 2 ) 2 = 2(|z| 2 + |w| 2 ) 2 -M(2|w| 2 + |z| 2 (1 + |w| 2 ) 2 ) (1 + |w| 2 ) 2 (|z| 2 + |w| 2 ) 2 < 0
for M > 2, z = 1 and arbitrary w. Therefore, the conditions of Theorem 3.1 do not hold.

3.1.

Weight functions ψ of special form. In this subsection we evaluate the dimension of F 2 ψ and the applicability of our criterion in Theorem 3.1, for some concrete weight functions ψ and for ψ in some special classes.

Example 3.5. Let k ≥ 3. Set ψ(z) = |z k 1 + z k 2 | 2 , z = (z 1 , z 2 ) ∈ C 2 . Given M > 0, we have L ψ-ψ M (z) = k 2 |z 1 | 2(k-1) -M |z| 4 |z 2 | 2 k 2 (z 1 z 2 ) k-1 + M |z| 4 z 1 z 2 k 2 (z 1 z 2 ) k-1 + M |z| 4 z 1 z 2 k 2 |z 2 | 2(k-1) -M |z| 4 |z 1 | 2 ,
and, hence,

det(L ψ-ψ M (z)) = k 2 |z 1 | 2(k-1) - M |z| 4 |z 2 | 2 k 2 |z 2 | 2(k-1) - M |z| 4 |z 1 | 2 -k 2 (z 1 z 2 ) k-1 + M |z| 4 z 1 z 2 k 2 (z 1 z 2 ) k-1 + M |z| 4 z 1 z 2 = - k 2 M |z| 4 |z 1 | 2k + |z 2 | 2k + (z 1 z 2 ) k + (z 1 z 2 ) k = - k 2 M |z| 4 |z k 1 + z k 2 | 2 < 0 when z k 1 + z k 2 = 0. Thus, for M > 0, the function ψ -ψ M is not plurisubharmonic outside a compact subset of C 2 .
Next we are going to verify that dim F 2 ψ = ∞. We have

X := C 2 e -|z k 1 +z k 2 | 2 dv(z) ≍ ∞ 0 S 2 r 3 e -r 2k |ζ k 1 +ζ k 2 | 2 dσ(ζ 1 , ζ 2 ) dr ≍ S 2 |ζ k 1 + ζ k 2 | -4/k dσ(ζ 1 , ζ 2 ).
Given ε > 0, we consider the set

T ε = (ζ 1 , ζ 2 ) ∈ S 2 : |ζ k 1 + ζ k 2 | < ε . Given (ζ 1 , ζ 2 ) ∈ S 2 such that |ζ 1 | ≥ |ζ 2 |, set ζ 1 = 1 2 + r • e iθ and ζ 2 = 1 2 -r • e iϕ , r ≥ 0. If (ζ 1 , ζ 2 ) ∈ T ε , then |ζ 1 | 2 -|ζ 2 | 2 < Cε for some constant C = C(k) > 0. Hence, r ε. Next, since |ζ k 1 + ζ k 2 | < ε, we obtain that |e ikθ -e ikϕ | ε. As a result, we obtain that σ(T ε ) ε 2 . Set U s = (ζ 1 , ζ 2 ) ∈ S 2 : 2 -s < |ζ k 1 + ζ k 2 | ≤ 2 -s+1 . Then X ≍ ∞ s=0 Us |ζ k 1 + ζ k 2 | -4/k dσ(ζ 1 , ζ 2 ) ∞ s=0 2 -2s 2 4s/k = ∞ s=0 2 -2s(1-(2/k)) < ∞, since k ≥ 3. Thus, 1 ∈ F 2 ψ .
In the same way, for every α > 0 we get

C 2 e -α|z k 1 +z k 2 | 2 dv(z) < ∞.
Consider the entire functions f (z) = e β(z k

1 +z k 2 ) 2 , 0 < β < 1 2 . Since C 2 e β(z k 1 +z k 2 ) 2 2 e -|z k 1 +z k 2 | 2 dv(z) = C 2 e 2β Re((z k 1 +z k 2 ) 2 )-|z k 1 +z k 2 | 2 dv(z) ≤ C 2 e -(1-2β)|z k 1 +z k 2 | 2 dv(z) < ∞, we conclude that dim F 2 ψ = ∞. Interestingly, F 2 ψ = 0 if k = 2. Indeed, let ψ((z 1 , z 2 )) = |z 2 1 + z 2 2 | 2 , f ∈ F 2 ψ , f (z 1 , z 2 ) = (z 2 1 + z 2 2 ) s g(z 1 , z 2 ) for some s ≥ 0, where g(z 1 , z 2 ) is not a multiple of z 2 1 + z 2 2
. By the mean value property, for every

z 1 ∈ C \ D(0, 10) we have |g(z 1 , iz 1 )| 2 (1 + |z 1 |) 2 D(iz 1 ,2/(1+|z 1 |)\D(iz 1 ,1/(1+|z 1 |)) |g(z 1 , z 2 )| 2 e -|z 2 1 +z 2 2 | 2 dv(z 2 ) (1 + |z 1 |) 2 D(iz 1 ,2/(1+|z 1 |)\D(iz 1 ,1/(1+|z 1 |)) |f (z 1 , z 2 )| 2 e -|z 2 1 +z 2 2 | 2 dv(z 2 ). Hence, C |g(z 1 , iz 1 )| 2 (1 + |z 1 |) -2 dv(z 1 ) f 2 ψ ,
and by a Liouville type theorem, g(z, iz) ≡ 0. Analogously, g(z, -iz) ≡ 0. Set h(z, w) = g(z -iw, z + iw). Then h is an entire function and h(0, w) = h(w, 0) ≡ 0. Hence, h(z, w) = zwh 1 (z, w) for another entire function h 1 and g(z 1 , z 2 ) = (z 2 1 + z 2 2 )g 1 (z 1 , z 2 ) for some entire function g 1 . This contradiction shows that F 2 ψ = 0. Extending the previous example to C n with n ≥ 3 requires a bit more work.

Example 3.6. Let n ≥ 3, k ≥ n + 1. Set ψ(z) = |z k 1 + • • • + z k n | 2 , z = (z 1 , . . . , z n ) ∈ C n .
Let us verify that for M > 0, the function ψ -ψ M is not plurisubharmonic outside a compact subset of C n .

We have

L ψ (z) = k 2      |z 1 | 2(k-1) (z 1 z 2 ) k-1 . . . (z 1 z n ) k-1 (z 1 z 2 ) k-1 |z 2 | 2(k-1) . . . (z 2 z n ) k-1 . . . . . . . . . . . . (z 1 z n ) k-1 (z 2 z n ) k-1 . . . |z n | 2(k-1)      = k 2      z k-1 1 z k-1 2 . . . z k-1 n      z 1 k-1 z 2 k-1 . . . z n k-1 . Set A(z) = M |z| 4     z 1 z 2 . . . z n     z 1 z 2 . . . z n .
Then

L ψ-ψ M (z) = L ψ (z) + A(z) - M |z| 2 I.
The spectra of the matrices L ψ (z) and A(z) are

σ L ψ (z) = k 2 |z 1 | 2(k-1) + |z 2 | 2(k-1) + • • • + |z n | 2(k-1) , 0 , σ A(z) = M |z| 2 , 0 .
Let V be the a unit vector in C n orthogonal to

     z k-1 1 z k-1 2 . . . z k-1 n      and to     z 1 z 2 . . . z n     . Then L ψ-ψ M (z)V, V = L ψ (z)V + A(z)V - M |z| 2 V, V = - M |z| 2 < 0.
Thus, for M > 0, the function ψ -ψ M is plurisubharmonic at no points of C n \ {0}.

Suppose that ψ(z) = ϕ(|z| 2 ) is a radial plurisubharmonic function of class C 2 . By the computations in Remark 3.2,

(3.2) ∂ 2 ψ ∂z j ∂ zk (z) = ϕ ′′ (|z| 2 ) zj z k + ϕ ′ (|z| 2 )δ jk .
The action of the Monge-Ampère operator on ψ is

(dd c ψ) n = 4n! det ∂ 2 ψ ∂z j ∂ zk dv = 4n!(ϕ ′ (|z| 2 )) n-1 (ϕ ′ (|z| 2 ) + |z| 2 ϕ ′′ (|z| 2 )) dv. Proposition 3.7. Suppose that ψ(z) = ϕ(|z| 2 ) is a radial plurisubhar- monic function of class C 2 . Then dim F 2 ψ = ∞ if and only if (3.3) C n (dd c ψ) n = ∞.
Proof. Since the spectrum of the matrix (3.2) consists of the eigenvalues ϕ ′ (|z| 2 ) and ϕ ′ (|z| 2 ) + |z| 2 ϕ ′′ (|z| 2 ), the first eigenvalue has multiplicity n-1 and the second one has multiplicity 1, we have ϕ ′ ≥ 0, (rϕ ′ (r)) ′ ≥ 0 on R + . Furthermore, we have

C n (dd c ψ) n = C ∞ 0 (ϕ ′ (r 2 )) n-1 (ϕ ′ (r 2 ) + r 2 ϕ ′′ (r 2 )) dr 2n = C ∞ 0 d (rϕ ′ (r)) n .
Thus, (3.3) is equivalent to the relation lim r→∞ rϕ ′ (r) = ∞. Now, if rϕ ′ (r) is bounded on R + , then ψ(z) = O(log |z|), |z| → ∞, and a version of the Liouville theorem shows that dim F 2 ψ < ∞. On the other hand, if lim r→∞ rϕ ′ (r) = ∞, then log |z| = o(ψ(z)), |z| → ∞, and the polynomials belong to F 2 ψ . Hence, dim F 2 ψ = ∞. For general C 2 plurisubharmonic functions, the radial case suggests the following question. Is it true that dim F 2 ψ = ∞ if and only if (3.3) holds? Our last example gives a negative answer to this question.

Example 3.8. Given subharmonic functions ψ j on the complex plane, 1 ≤ j ≤ n, set (3.4) ψ(z 1 , . . . , z n ) = n j=1 ψ j (z j ).

Claim: dim F 2 ψ < ∞ if and only if either max j dim F 2 ψ j < ∞ or min j dim F 2 ψ j = 0.

In one direction, by the Fubini theorem, if dim F 2 ψ < ∞, then max j dim F 2 ψ j < ∞ or min j dim F 2 ψ j = 0. In the opposite direction, it is clear that if min j dim F 2 ψ j = 0, then F 2 ψ = 0. It remains to verify that if max j dim F 2 ψ j < ∞, then dim F 2 ψ < ∞. First, suppose that n = 2, dim In the same way, the functions f (•, w j ), 1 ≤ j ≤ N, belong to F 2 Hence, every a j belongs to F 2 ψ 1 . Since dim F 2 ψ 1 < ∞, we conclude that the space F 2 ψ has finite dimension. For n ≥ 2 we can just use an inductive argument. This completes the proof of Claim.

F 2 ψ 1 < ∞, N = dim F 2 ψ 2 < ∞.
Let us return to general ψ satisfying (3.4). We have

  Fix a basis (g k ), 1 ≤ k ≤ N, in the space F 2 ψ 2 and choose a family of points(w m ), 1 ≤ m ≤ N, such that det Q = 0, where Q = g k (w m ) N k,m=1 . Next, choose f ∈ F 2 ψ .By the mean value property,|f (z, w)| 2 ≤ 1 π D(z,1) |f (ζ, w| 2 dv(ζ), z, w ∈ C.Therefore, for every z ∈ C, the function f (z, •) belongs to F 2 ψ 2 , and, hence, we havef (z, •) = N k=1 a k (z)g k .

(

  dd c ψ) n = C C n n j=1 ∆ψ j (z j ) dv(z) = C n j=1 C ∆ψ j (z j ) dv(z j ). Now, if n = 2, ψ 1 (z) = |z| 2 , ∆ψ 2 (z) = max(1 -|z|, 0), then C n (dd c ψ) n = ∞, but F 2 ψ = 0.Thus, Proposition 3.7 does not extend to general C 2smooth plurisubharmonic functions.

Acknowledgments. We thank Friedrich Haslinger and Grigori Rozenblum for helpful remarks.

The results of Section 2 were obtained in the framework of the project 20-61-46016 by the Russian Science Foundation. A. Borichev and H. Youssfi were partially supported by the project ANR-18-CE40-0035.

Finally, let us verify that dim F 2 ψ = ∞. Set

Given ε > 0, we consider the set

Then the function f = log |P | is plurisubharmonic. Following [START_REF] Ch | Ensembles de sous-niveau et images inverses des fonctions plurisousharmoniques[END_REF], we consider the Lelong number of

If f (a) = 0, then ν f (a) = 0. Otherwise, let a = (a 1 , . . . , a n ) = 0 and f (a) = 0. Without loss of generality, we can assume that a

By homogeneity of P ,

for some constant C > 0. Arguing as in Example 3.5, we obtain first that 1 ∈ F 2 ψ and then that dim F 2 ψ = ∞ for k ≥ n + 1. At the end of the paper, we consider two special classes of weight functions ψ: radial weight functions and the functions of the form ψ(z 1 , . . . , z n ) = n j=1 ψ j (z j ).


Zagrebnov

Approximations of self-adjoint C 0 -semigroups in the operator-norm topology Dedicated to Yuri Kozitsky on the occasion of

Keywords: 2010 Mathematics Subject Classification. 47D05, 47A55, 81Q15, 47B65 Strongly continuous semigroup, Chernoff approximation formula, Trotter-Kato product formulae

VALENTIN A

Introduction

The aim of the paper is to present a new generalised proof of approximation theory developed 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]. For self-adjoint Trotter-Kato product formulae it allows to obtain optimal estimate for the rate of convergence in operator norm for Kato functions of class K β , where β = 2 (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]).

Instead of a double-iteration procedure 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] we extend in this paper the Chernoff approximation formula [START_REF] Chernoff | Note on product formulas for operator semigroups[END_REF] and the Trotter-Neveu-Kato approximation theorem [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF], Theorem IX.2.16, to the operator-norm topology. Essentially we follow here the idea of lifting the strongly convergent Chernoff approximation formula to operator-norm convergence [START_REF] Neidhardt | Trotter-Kato product formula and symmetrically normed ideals[END_REF][START_REF] Zagrebnov | Comments on the Chernoff √ n-lemma[END_REF], whereas majority of results concerning this formula are about the strong operator topology, see, for example, review [START_REF] Butko | The method of Chernoff approximation[END_REF]. In the same vein we quote a recent book [START_REF] Bobrowski | Convergence of One-parameter Operator Semigroups[END_REF], where different aspects of semigroup convergence in the strong operator topology are presented in great details.

To proceed, we first recall definition of the Kato functions that belong to the class K β .

Definition 1.1. If a real-valued Borel measurable function

f : [0, ∞) → [0, 1] satisfying (1.1) 0 ≤ f (s) ≤ 1, f (0) = 1, f (+0) = -1,
is such that for any ε > 0 there exists a positive constant δ ε < 1 implying

(1.2) f (s) ≤ 1 -δ ε , s ≥ ε,
and that for some β, where 1 < β ≤ 2,

(1.3) [f ] β := sup s>0 |f (s) -1 + s| s β < ∞, then f ∈ K β .
Some elementary examples of functions satisfying Definition 1.1 are

(1.4) f (s) = e -s , f (s) = (1 + k -1 s) -k , k > 0.
Note that the Kato functions of class K β are not necessarily monotonously decreasing, but it is true in a vicinity of s = +0. For more details about different types of Kato functions see Appendix C in [START_REF] Zagrebnov | Gibbs Semigroups[END_REF]. By Definition 1.1 and by the spectral theorem one gets that for any non-negative self-adjoint operator A the bounded operator-valued function t → f (tA) ∈ L(H) is strongly continuous in R + and right-continuous on R + 0 = R + ∪ {0}, that is, s-lim t→+0 f (tA) = 1.

One of the main corollaries of the semigroup approximation results established in the present paper (Theorem 4.5) is the statement about operator norm convergence of the Trotter-Kato product formulae, see Section 5. Proposition 1.2. Let f, g ∈ K 2 . If A and B are non-negative self-adjoint operators in a separable Hilbert space H with domains dom A and dom B such that the operator sum

C := A + B is self-adjoint on dom C = dom A ∩ dom B, then g(tB/n) 1/2 f (tA/n)g(tB/n) 1/2 n -e -tC = O(n -1 ), (1.5) f (tA/n)g(tB/n) n -e -tC = O(n -1 ), (1.6)
for n → ∞, hold in the operator norm topology. The convergence is locally uniform on R + 0 , but if operator C is strictly positive, it is uniform on R + 0 . Note that the rates of convergence in (1.5) and in (1.6) are optimal, i.e., they can not be improved in the general setup [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].

Chernoff approximation formula: strong operator topology

In this section we give a proof of the Chernoff approximation formula in the strong operator topology, which is alternative to the original one based on the √ n-Lemma [START_REF] Chernoff | Note on product formulas for operator semigroups[END_REF]. In conclusion we relax some conditions of the main Theorem 2.3.

Let F (•) : R + 0 -→ L(H) be a measurable family of non-negative selfadjoint contractions

F (t) ≤ 1 such that F (0) = 1. We set (2.1) S(τ ) := 1 -F (τ ) τ , τ > 0.
Then for each τ > 0 bounded operator S(τ ) is self-adjoint and positive.

Let H ≥ 0 be self-adjoint operator in H. Then by the Trotter-Neveu-Kato convergence theorem we obtain that

(2.2) s-lim τ →+0 (1 + S(τ )) -1 = (1 + H) -1 ,
holds if and only if

(2.3) s-lim τ →+0 e -tS(τ ) = e -tH ,
uniformly in t ∈ I for any closed bounded interval I ⊂ R + 0 . In this case we say that this convergence holds locally uniformly in t ∈ R + 0 , whereas if I ⊂ R + , then convergence holds locally uniformly away from zero. For example, setting τ = t/η for η ≥ 1, we obtain (2.4) slim η→+∞ e -tS(t/η) = e -tH , locally uniformly away from zero.

To proceed, we need the following elementary estimate:

Lemma 2.1. For λ ∈ [0, 1] and r ≥ 1, one has

(2.5) 0 ≤ e -r(1-λ) -λ r ≤ 1 r .
The next assertion serves to lift the weak convergence of vectors {u n } n≥1 in H to the strong convergence of this sequence.

F (t/η) η -e -tS(t/η) = 0,
locally uniformly in t > 0. To this end we use the spectral functional calculus for self-adjoint operators to obtain the representation

(2.8) F (τ ) r -e -r(1-F (τ )) = [0,1] dE F (τ ) (λ) λ r -e -r(1-λ) , τ > 0, r ≥ 1.
Then inequality (2.5) yields the estimate

F (τ ) r -e -r(1-F (τ )) u ≤ r -1 u , r ≥ 1, u ∈ H.
Setting τ = t/η, η ≥ 1, and r = t/τ ≥ 1, we obtain

F (t/η) η -e -tS(t/η) u ≤ η -1 u , η ≥ 1, u ∈ H.
F (t/n) n = e -tH ,
also holds locally uniformly away from zero for the sequence when η = n ∈ N.

We note that formula (2.9) follows from general Banach space Chernoff approximation formula, see Theorem 2.2 in [START_REF] Zagrebnov | Comments on the Chernoff √ n-lemma[END_REF], as a particular case for self-adjoint semigroups. Lemma 2.4. Let K(•) : R + → L(H) be a measurable family of non-negative self-adjoint operators and let H be a non-negative self-adjoint operator. If the weak operator limit:

(2.10) w-lim τ →+0 (λ1 + K(τ )) -1 = (λ1 + H) -1 ,
for each λ > 0, then it is also true in the strong operator topology:

(2.11) s-lim τ →+0 (λ1 + K(τ )) -1 = (λ1 + H) -1 .
Proof. By virtue of (2.10) we get (2.12)

lim τ →+0 (λ1 + K(τ )) -1/2 u = (λ1 + H) -1/2 u , u ∈ H, for λ > 0. Since (λ 1 + K(τ )) -1/2 = 1 π ∞ 0 dx 1 √ x (x 1 + λ 1 + K(τ )) -1 , the limit (2.10) yields w-lim τ →+0 (λ 1 + K(τ )) -1/2 = (λ 1 + H) -1/2
. This, together with (2.12) and the lifting Proposition 2.2, imply s-lim τ →+0 (λ

1 + K(τ )) -1/2 = (λ 1 + H) -1/2
. Since the product of operators is strongly continuous, this limit yields (2.11).

Taking into account Lemma 2.4, the conditions of Theorem 2.3 can be relaxed as follows.

Theorem 2.5. Let F (•) : R + 0 -→ L(H) be a measurable family of nonnegative contractions such that F (0) = 1 and let H ≥ 0 be a self-adjoint operator in H. The statement (2.6) is valid if and only if for each λ > 0 the condition

(2.13) w-lim τ →+0 (λ1 + S(τ )) -1 = (λ1 + H) -1 , is satisfied.
We skip the proof since the line of reasoning is straightforward.

Lifting the Chernoff approximation formula to operator-norm topology

A natural question arises: can the limit (2.6) in Theorem 2.3 (or in Theorem 2.5) be lifted to convergence in the operator-norm topology? First we note that in contrast to quasi-sectorial contractions [START_REF] Zagrebnov | Quasi-sectorial contractions[END_REF], the estimates for self-adjoint contraction C in the Chernoff √ n-Lemma [START_REF] Chernoff | Note on product formulas for operator semigroups[END_REF] and in its refinement due to the 1/ 3 √ n -Theorem (see Lemma 2.1 and Theorem 3.3 in [START_REF] Zagrebnov | Comments on the Chernoff √ n-lemma[END_REF]) can be significantly improved. Namely, the spectral functional calculus of self-adjoint contraction C and Lemma 2.1 yield

(3.1) C n -e n(C-1) = 1 0 dE C (λ) λ n -e n(λ-1) ≤ 1 n , n ∈ N.
Similarly to the case of the strong operator topology, the next step in the program of lifting the approximation formula to operator-norm topology involves the lifting of the Trotter-Neveu-Kato convergence theorem. Therefore, we proceed with the following lemma, which is well suited for selfadjoint lifting of this theorem.

Lemma 3.1. Let K and L be non-negative self-adjoint operators in a Hilbert space H. Then

(3.2) e -K -e -L ≤ c (1 + K) -1 -(1 + L) -1
with a constant c > 0 independent of operators K and L.

Proof. By the Riesz-Dunford functional calculus, one obtains for the difference of exponentials the representation

(3.3) e -K -e -L = 1 2πi Γ dz e -z (z -K) -1 -(z -L) -1 ,
where the contour Γ is a union of two branches:

Γ = Γ 0 ∪ Γ ∞ , with Γ 0 = {z ∈ C : z = e iϕ , π/4 ≤ ϕ ≤ 2π -π/4}, Γ ∞ = {z ∈ C : z = re ±iπ/4 , r ≥ 1}. (3.4) From (3.3) we find the representation e -K -e -L = 1 2πi Γ dz e -z (1 + K)(z -K) -1 × × (1 + L) -1 -(1 + K) -1 (1 + L)(z -L) -1 .
(3.5)

Since (1 + K)(z -K) -1 = -1 + (1 + z)(z -K) -1
, one gets the estimate

(1 + K)(z -K) -1 ≤ 1 + 1 + |z| dist (z, R + ) . Setting c Γ := sup z∈Γ 1 + |z| dist (z, R + ) < ∞, we find (3.6) sup z∈Γ (1 + K)(z -K) -1 ≤ (1 + c Γ ),
where the constant c Γ depends only on Γ but not on the operator K. Similarly, from (3.6) one also gets

sup z∈Γ (1 + L)(z -L) -1 ≤ (1 + c Γ ).
Using these estimates, we find from (3.5) that

e -K -e -L ≤ c (1 + K) -1 -(1 + L) -1 with c := 1 2π (1 + c Γ ) 2 Γ |dz| |e -z |.
Since for z ∈ Γ ∞ the value of e z > 0, the integral is convergent and c depends only on the contour Γ.

The first step towards the proof the operator-norm convergence of the Chernoff approximation formula (2.6) would be lifting of the strong convergence in (2.2) to the operator-norm convergence. To study the consequence of this lifting we prove the following assertion. Lemma 3.2. Let F (•) : R + 0 -→ L(H) be a measurable family of nonnegative self-adjoint contractions such that F (0) = 1. Let self-adjoint family {S(τ )} τ >0 be defined by (2.1)-(2.2), where H is a non-negative self-adjoint operator in H. Then the condition

(3.7) lim τ →+0 (1 + S(τ )) -1 -(1 + H) -1 = 0,
is satisfied if and only if

(3.8) lim η→∞ sup t∈I (1 + tS(t/η)) -1 -(1 + tH) -1 = 0,
for any closed interval I ⊂ R + , i.e., locally uniformly away from zero.

Proof. A straightforward computation shows that

(1 + tS(τ )) -1 -(1 + tH) -1 = t(1 + S(τ ))(1 + tS(τ )) -1 [(1 + S(τ )) -1 -(1 + H) -1 ](1 + H)(1 + tH) -1 .
Here we used the fact that if t > 0 and τ > 0, then for self-adjoint operator S(τ ) the closure

(1 + tS(τ )) -1 (1 + S(τ )) = (1 + S(τ ))(1 + tS(τ )) -1 .
For these values of arguments t and τ we get

(1 + S(τ ))(1 + tS(τ )) -1 ≤ (1 + 2/t), (1 + H)(1 + tH) -1 ≤ (1 + 2/t). If I is a closed interval of R + , for example, I := [a, b], 0 < a < b < ∞, then (1 + tS(τ )) -1 -(1 + tH) -1 ≤ b(1 + 2/a) 2 (1 + S(τ )) -1 -(1 + H) -1 ,
for t ∈ I and τ > 0. Setting τ = t/η we find (3.9)

(1 + tS(t/η)) -1 -(1 + tH) -1 ≤ b(1 + 2/a) 2 (1 + S(t/η)) -1 -(1 + H) -1 .
Since by (3.7) we obtain for the last factor in the right-hand side of (3.9)

lim η→∞ sup t∈I (1 + S(t/η)) -1 -(1 + H) -1 = 0,
the estimate (3.9) yields (3.8). The converse is obvious. Proof. For t > 0 and η > 0, we obviously have the estimate

(3.11) F (t/η) η -e -tH ≤ F (t/η) η -e -tS(t/η) + e -tS(t/η) -e -tH .
By the functional calculus of self-adjoint contraction C := F (τ ), Lemma 2.1 yields estimate (3.1), which improves the Chernoff √ n-Lemma [START_REF] Chernoff | Note on product formulas for operator semigroups[END_REF], as well as an estimate in [START_REF] Zagrebnov | Comments on the Chernoff √ n-lemma[END_REF]. Then (3.1), for continuous variable n = η, and (3.11), where S(t/η) is defined by (2.1), imply (3.12) F (t/η) η -e -tH ≤ 1 η + e -tS(t/η) -e -tH , t > 0, η > 0 .

Note that by Lemma 3.1, there is a constant c > 0 such that

(3.13) e -tS(t/η) -e -tH ≤ c (1 + tS(t/η)) -1 -(1 + tH) -1 ,
for t > 0, η > 0. Inserting the estimate (3.13) into (3.12), we obtain

(3.14) F (t/η) η -e -tH ≤ 1 η + c (1 + tS(t/η)) -1 -(1 + tH) -1 ,
Then applying Lemma 3.2, we get (3.10).

Conversely, let us assume (3.10). Note that e -tS(t/η) -e -tH ≤ F (t/η) η -e -tH + F (t/η) η -e -tS(t/η) , for t > 0 and η > 0. Applying to the last term the spectral representation for F (t) and Lemma 2.1 for the corresponding integrand, we find for t > 0 and η > 0

(3.15) e -tS(t/η) -e -tH ≤ F (t/η) η -e -tH + 1 η .
Then by assumption (3.10) the estimate (3.15) yields (3.16) lim η→∞ sup t∈I e -tS(t/η) -e -tH = 0, locally uniformly away from zero. Hence, the limit lim η→∞ e -tS(t/η)e -tH = 0, or equivalently lim τ →+0 e -tS(τ ) -e -tH = 0, holds for any t > 0. Now, using representation:

(3.17)

(1 + S(τ )) -1 -(1 + H) -1 = ∞ 0 ds e -s e -sS(τ ) -e -sH ,
we obtain the estimate

(3.18) (1 + S(τ )) -1 -(1 + H) -1 ≤ ∞ 0 ds e -s e -sS(τ ) -e -sH .
Let Φ τ (s) := e -s e -sS(τ ) -e -sH . Since S(τ ) ≥ 0 and H ≥ 0, one gets Φ τ (s) ≤ 2 e -s ∈ L 1 (R + 0 ) and lim τ →+0 Φ τ (s) = 0. Then limit lim τ →+0 in the right-hand side of (3.18) is zero by the Lebesgue dominated convergence theorem, that yields (3.7). Lemma 3.4. Let {X n } n>0 be a sequence of bounded non-negative selfadjoint operators such that lim n→∞ X n -X = 0 for a linear operator X.

Then (i) X ∈ L(H) and X = X * ≥ 0; (ii) for any continuous function g(•) : [0, X ] -→ R one gets lim n→∞ g(X n ) -g(X) = 0.
Proof. (i) This part is a straightforward corollary of the properties of the sequence {X n } n≥1 . (ii) Note that lim n→∞ X n -X = 0 implies lim n→∞ X n n -X n = 0 for n ∈ N, and estimate X n ≤ X + δ for any δ > 0, where n > N (δ). Then

lim n→∞ p(X n ) -p(X) = 0, for any polynomial p : [0, X ] -→ R.
By the Weierstrass theorem, polynomials are dense in the set of continuous functions C ∞ ([0, X ]) in topology • ∞ of uniform convergence. Thus, for any given ε > 0, we can find polynomial p(•) such that g -p ∞ = sup x∈[0, X ] |g(x) -p(x)| < ε/3. Then by spectral representation for operators X and X n , we obtain for n > N (ε):

g(X) -p(X) < ε/3 , g(X n ) -p(X n ) < ε/3 . Now taking n > N (δ) ∧ N (ε), one gets p(X n ) -p(X) < ε/3,
which consequently yields the estimate g(X n ) -g(X) < ε and therefore proves the lemma. Proof. We use the representation

(3.20) F (t/η) η = (F (t 0 /ν) ν ) t/t 0 , ν := t 0 t η, t > 0.
Let X := e -t 0 H and X ν := F (t 0 /ν) ν . Then by assumption (3.19), lim ν→∞ X ν -X = 0. Now, let function x → g(x) := x t/t 0 be defined for x ≥ 0. Then by Lemma 3.4, we obtain

lim ν→∞ X t/t 0 ν -X t/t 0 = 0 ,
and by virtue of representation (3.20) it follows that lim η→∞ F (t/η) ηe -tH = 0 holds for any t ∈ R + . Now, proceeding as above in the proof of Theorem 3.3, one deduces (3.18), which yields (3.7). Finally, applying Theorem 3.3, we obtain (3.10) for any closed interval I ⊂ R + .

Since by definition of {e -sS(τ ) } s≥0 and by C 0 -semigroup property of {e -sH } t≥0 the corresponding strong limits: s-lim s→+0 , are well-defined, the lim τ →+0 e -sS(τ ) -e -sH = 0 in (3.18) is valid also for s = 0. A question arises: what happens if the condition (3.16) is satisfied uniformly for any bounded interval I ⊂ R + 0 ? Theorem 3.6. Let F (•) : R + 0 -→ L(H) be a measurable family of nonnegative self-adjoint contractions such that F (0) = 1. Let self-adjoint family {S(τ )} τ >0 be defined by (2.1)-(2.2), where H is non-negative self-adjoint operator in H. Then the convergence Conversely, assume (3.21). Note that by representation (3.17), one gets

(1 + tS(τ )) -1 -(1 + tH) -1 = ∞ 0
ds e -s e -s tS(τ ) -e -s tH , t ≥ 0 , that yields the estimate

(1 + tS(τ )) -1 -(1 + tH) -1 ≤ ∞ 0 ds e -s e -s tS(τ ) -e -s tH ,
for τ > 0 and t ≥ 0. Now, let 0 < ε < 1 and let N ε := -ln(ε/2). Then ∞ Nε ds e -s e -s tS(τ ) -e -s tH ≤ ε, for τ > 0 and t ≥ 0. Hence,

(1 + tS(τ )) -1 -(1 + tH) -1 ≤ Nε 0 ds e -s e -stS(τ ) -e -stH + ε, that yields sup t∈I (1 + tS(τ )) -1 -(1 + tH) -1 ≤ sup t ∈ I s ∈ [0, N ε ] e -s tS(τ ) -e -s tH + ε,

Operator-norm approximation and estimates of the rate of convergence

Theorem 3.7 admits further modifications. In particular, it allows establishing estimates for the rate of operator-norm convergence.

Theorem 4.1. Let F (•) : R + 0 -→ L(H) be a measurable family of nonnegative self-adjoint contractions such that F (0) = 1. Let self-adjoint family {S(τ )} τ >0 be defined by (2.1)-(2.2), where H is non-negative self-adjoint operator in H. (i) If ρ ∈ (0, 1] and there is a constant M ρ > 0 such that the estimate

(4.1) (1 + tS(τ )) -1 -(1 + tH) -1 ≤ M ρ τ t ρ ,
holds for τ, t ∈ (0, 1] and 0 < τ ≤ t, then there is a constant c ρ > 0 such that the estimate

(4.2) F (τ ) t/τ -e -tH ≤ c ρ τ t ρ , is valid for 0 < τ ≤ t ≤ 1.
(ii) If ρ ∈ (0, 1) and there is a constant c ρ such that (4.2) holds, then there is a constant M ρ > 0 such that the estimate (4.1) is valid for 0 < τ ≤ t ≤ 1.

Proof. (i) By Lemma 3.1, there is a constant c > 0 such that

(4.3) e -tS(τ ) -e -tH ≤ c (1 + tS(τ )) -1 -(1 + tH) -1 ,
for τ, t > 0. Using (4.1), we obtain

e -tS(τ ) -e -tH ≤ c M ρ τ t ρ .
If 0 < τ ≤ t, the inequality (2.5) and the spectral representation for F (τ ) (2.8) yield (4.4)

F (τ ) t/τ -e -tS(τ ) ≤ τ t .
Then estimate

(4.5) F (τ ) t/τ -e -tH ≤ F (τ ) t/τ -e -tS(τ ) + e -tS(τ ) -e -tH gives F (τ ) t/τ -e -tH ≤ τ t + c M ρ τ t ρ , for τ, t ∈ (0, 1] with 0 < τ ≤ t. Since for ρ ∈ (0, 1] one has τ /t ≤ (τ /t) ρ , this implies (4.6) F (τ ) t/τ -e -tH ≤ (1 + c M ρ ) τ t ρ .
Setting c ρ := 1 + c M ρ , we prove (4.2) for ρ ∈ (0, 1].

(ii) To prove (4.1), we use the representation

(1 + tS(τ )) -1 -(1 + tH) -1 = ∞ 0 dx e -x e -xtS(τ ) -e -xtH ,
for τ, t > 0. Then we get

(1 + tS(τ )) -1 -(1 + tH) -1 = ∞ n=0 n+1 n dx e -x e -xtS(τ ) -e -xtH .
Substitution x = y + n yields

(1 + tS(τ )) -1 -(1 + tH) -1 = ∞ n=0 e -n 1 0
dy e -y e -(y+n)tS(τ ) -e -(y+n)tH .

Since

e -(y+n) t S(τ ) -e -(y+n) t H = e -n t S(τ ) -e -n t H e -y t S(τ ) + e -n t H e -y t S(τ ) -e -y t H , and e -n tS(τ ) -e -n tH = n-1 k=0 e -ktS(τ ) e -tS(τ ) -e -tH e -(n-k-1)tH , we get

(1 + tS(τ )) -1 -(1 + tH) -1 = ∞ n=0 e -n n-1 k=0
e -ktS(τ ) e -tS(τ ) -e -tH e -(n-k-1)tH 1 0 dy e -y e -ytS(τ ) + e -ntH 1 0 dy e -y e -ytS(τ ) -e -ytH .

Hence, we obtain for τ, t > 0 the estimate (4.7)

(1 + tS(τ )) -1 -(1 + tH) -1 ≤ ∞ n=0
e -n n e -tS(τ ) -e -tH + 1 0 dy e -y e -ytS(τ ) -e -ytH .

Note that assumption (4.2) and estimate (4.4) yield

(4.8) e -tS(τ ) -e -tH ≤ (1 + c ρ ) τ t ρ , for 0 < τ ≤ t ≤ 1.
To treat the last term in (4.7) we use decomposition Taking into account (4.10) and (4.11), we obtain from (4.9) the estimate (4.12)

1 0 dy e -y e -ytS(τ ) -e -ytH ≤ (1+c ρ ) 1 0 dy e -y y -ρ +2 τ t ρ .
Finally, using (4.8) and (4.12), one gets for (4.7) the estimate

(1 + tS(τ )) -1 -(1 + tH) -1 ≤ ∞ n=0 e -n n (1 + c ρ ) + (1 + c ρ ) 1 0 dy e -y y -ρ + 2 τ t ρ . Now setting (4.13) M ρ := ∞ n=0 e -n n (1 + c ρ ) + (1 + c ρ ) 1 0 dy e -y y -ρ + 2 ,
we obtain the estimate (4.1) for 0 < τ ≤ t ≤ 1.

In Theorem 4.1(i) it is shown that for ρ = 1 the condition (4.1) implies (4.2). Since integral in (4.13) diverges for ρ = 1, it is unclear whether the converse is also true. Hence, Theorem 4.1(ii) does not cover this case.

Note that the setting τ = t/η transforms inequality (4.2) into

F (t/η) η -e -tH ≤ c ρ 1 η ρ , η ≥ 1.
This inequality gives the convergence rate estimate for restricted interval: t ∈ (0, 1], and local conditions: 0 < τ ≤ t ≤ 1. The same conditions yield generalisation of (4.14) to any bounded interval I ⊂ R + 0 .

Theorem 4.2. Let F (•) : R + 0 -→ L(H) be a measurable family of nonnegative self-adjoint contractions such that F (0) = 1. Let self-adjoint family {S(τ )} τ >0 be defined by (2.1)- (2.2), where H is a non-negative self-adjoint operator in H.

If for some ρ ∈ (0, 1] there is a constant M ρ > 0 such that the estimate (4.1) holds for τ, t ∈ (0, 1] and 0 < τ ≤ t, then for any bounded interval I ⊂ R + 0 there is a constant c I ρ > 0 such that the estimate

F (t/N η) N η -e -t H = N -1 k=0 e -k t H/N (F (t/N η) η -e -t H/N )F (t/N η) (N -1-k)η ,
yields the estimate

F (t/N η) N η -e -t H ≤ N F (t/N η) η -e -t H/N .
Let t := t/N and τ := t /η, η ≥ 1. Then 0 < τ ≤ t ≤ 1. Applying Theorem 4.1, we find that

F (t/N η) η -e -tH/N = F (τ ) t /τ -e -t H ≤ c ρ τ t ρ .
This implies for 0 < τ ≤ t ≤ 1, i.e., for t ≤ N the estimate

F (t/N η) N η -e -tH ≤ c ρ N τ t ρ .
Since τ = t /η, then for η := N η ≥ 1 we get, cf. (4.14),

F (t/η ) η -e -tH ≤ c ρ N 1+ρ 1 η ρ , t ∈ [0, N ].
Setting c If for some ρ ∈ (0, 1] there is a constant M ρ > 0 such that the estimate (4.1) holds for 0 < τ ≤ t < ∞, then there is a constant c R + ρ > 0 such that the estimate (4.16)

sup t∈R + 0 F (t/η) η -e -tH ≤ c R + ρ 1 η ρ , holds for η ≥ 1.
Proof. The line of reasoning that leads from (4.3) to the estimate (4.6) is obviously still valid if we assume 0 < τ ≤ t < ∞. Then setting τ := t/η, we deduce from (4.6)

F (t/η) η -e -tH ≤ c R + ρ 1 η ρ , η ≥ 1,
where c R + ρ := 1 + c M ρ and t ∈ R + 0 . For the case ρ = 1 the assumption (4.1) can be simplified and reduced to t-independent canonical form (3.7). To use Theorem 4.1 and Theorem 4.2, we return to local conditions: 0 < τ ≤ t ≤ 1. Theorem 4.4. Let F (•) : R + 0 -→ L(H) be a measurable family of nonnegative self-adjoint contractions such that F (0) = 1. Let self-adjoint family {S(τ )} τ >0 be defined by (2.1)-(2.2), where H is a non-negative self-adjoint operator in H.

If there is a constant M 1 > 0 such that the estimate (4.17)

(1 + S(τ )) -1 -(1 + H) -1 ≤ M 1 τ ,
holds for τ ∈ (0, 1], then for any bounded interval I ⊂ R + 0 there is a constant c I 1 > 0 such that the estimate

(4.18) sup t∈I F (t/η) η -e -tH ≤ c I 1 1 η , holds for η ≥ 1.
Proof. For t > 0 the identity

(1 + tS(τ )) -1 -(1 + tH) -1 = t(1 + S(τ ))(1 + tS(τ )) -1 [(1 + S(τ )) -1 -(1 + H) -1 ](1 + H)(1 + tH) -1 , yields estimate (1 + tS(τ )) -1 -(1 + tH) -1 ≤ M 1 τ t (1 + S(τ ))(1 + tS(τ )) -1 (1 + H)(1 + tH) -1 , (4.19)
where we used condition (4.17).

Let 0 < t ≤ 1. Then (1 + S(τ ))(1 + tS(τ )) -1 ≤ 1/t and (1 + H)((1 + tH) -1 ≤ 1/t. Therefore (4.19) implies estimate (4.1) for 0 < τ ≤ t ≤ 1 and ρ = 1. By virtue of Theorem 4.1 we obtain (4.14). Finally, applying Theorem 4.2 for ρ = 1 we extend the proof of (4.18) to any bounded interval I ⊂ R + 0 . To extend Theorem 4.4 (case ρ = 1) to infinite interval I = R + 0 we add more conditions, including a global one. Theorem 4.5. Let in addition to conditions of Theorem 4.4 the operator H ≥ µ1, µ > 0. Moreover, we assume that for any ε ∈ (0, 1] there exists δ ε ∈ (0, 1) such that

(4.20) 0 ≤ F (τ ) ≤ (1 -δ ε )1, is valid for τ ≥ ε, cf. Definition 1.1.
If there is constant M 1 > 0 such that the (4.17) holds for τ ∈ (0, ε), then there exists constant c R + 1 > 0 such that estimate (4.18) is valid for infinite interval I = R + 0 . Proof. Since (4.17) implies the resolvent-norm convergence of {S(τ )} τ >0 , when τ → +0, and since H ≥ µ1, there exists 0 < µ ≤ µ such that S(τ ) ≥ µ 1 for τ ∈ (0, ε), where ε ≤ 1.

On the other hand, (4.19) yields that for t > 0

(1 + tS(τ )) -1 -(1 + tH) -1 = ≤ M 1 τ t (1 + S(τ ))(1/t + S(τ )) -1 (1 + H)(1/t + H) -1 . (4.21)
Since S(τ ) ≥ µ 1, τ ∈ (0, ε), and H ≥ µI, for t > 0 we obtain estimates for τ ∈ (0, ε) and 0 < t < ∞. Here M R + 1 := M 1 (1 + µ )(1 + µ)/µ µ. Note that if τ /t ≤ 1, then (4.23), for 0 < τ ≤ t < ∞ and τ ∈ (0, ε), satisfies conditions of Theorem 4.3. Indeed, for τ /t ≤ 1 and spectral representation for F (τ ) ≥ 0 we obtain the estimate F (τ ) t/τ -e -tS(τ ) ≤ τ t , 0 < τ ≤ t < ∞, which together with (4.3) for (4.23) and (4.5) allow to extend the result (4.18) of Theorem 4.4 to the case 0 < τ ≤ t < ∞, for τ = t/η, η ≥ 1. Since, τ = t/η < ε, this yields 2. To prove convergence of the sequences of non-self-adjoint approximants (1.6), we note that for n ∈ N and t ≥ 0:

(1 + S(τ ))(1/t + S(τ )) -1 ≤ 1 + µ µ , (1 + H)(1/t + H) -1 ≤ 1 + µ µ .
(f (tA/n)g(tB/n)) n = f (tA/n)g(tB/n) 1/2 F (t/n) n-1 g(tB/n) 1/2 .

Using the representation (f (tA/n)g(tB/n)) n -e -tC = f (tA/n)g(tB/n) 1/2 (F (t/n) n-1 -e -tC )g(tB/n) 1/2 + f (tA/n)g(tB/n) 1/2 e -tC (g(tB/n) 1/2 -1) + f (tA/n)(g(tB/n) 1/2 -1)e -tC + (f (tA/n) -1)e -tC , we obtain the following estimate:

(f (tA/n)g(tB/n)) n -e -tC ≤ F (t/n) n-1 -e -tC + 2 (1 -g(tB/n) 1/2 )e -tC + (1 -f (tA/n))e -tC .

Since (1 -g(tB/n) 1/2 )e -tC ≤ (1 -g(tB/n))e -tC , we obtain (f (tA/n)g(tB/n)) n -e -tC ≤ F (t/n) n-1 -e -tC + 2 (1 -g(tB/n))e -tC + (1 -f (tA/n))e -tC .

Lemma 3 .

 3 2 allows to advance in generalisation of self-adjoint Chernoff approximation formula for operator-norm convergence. Theorem 3.3. Let F (•) : R + 0 -→ L(H) be a measurable family of nonnegative self-adjoint contractions such that F (0) = 1. Let self-adjoint family {S(τ )} τ >0 be defined by (2.1)-(2.2), where H ≥ 0 is self-adjoint operator in H. Then we have (3.10) lim η→∞ sup t∈I F (t/η) η -e -tH = 0, for any closed interval I ⊂ R + , if and only if the family {S(τ )} τ >0 satisfies condition (3.7).

Corollary 3 . 5 .

 35 Let F (•) : R + 0 -→ L(H) be a measurable family of nonnegative self-adjoint contractions such that F (0) = 1. Let self-adjoint family {S(τ )} τ >0 be defined by (2.1)-(2.2), where H is non-negative self-adjoint operator in H. If(3.19) lim η→∞ F (t 0 /η) η -e -t 0 H = 0, holds for some t 0 > 0, then (3.10) holds for any closed interval I ⊂ R + .

( 1 +

 1 (τ ) -e -tH = 0, holds for any bounded interval I ⊂ R + 0 if and only if the condition(3.22) lim τ →0 sup t∈I tS(τ )) -1 -(1 + tH) -1 = 0,is valid for any bounded interval I ⊂ R + 0 . Proof. By conditions of theorem and by Lemma 3.1, we obtain from (3.13) the estimate sup t∈I e -tS(τ ) -e -tH ≤ c sup t∈I (1 + tS(τ )) -1 -(1 + tH) -1 , for τ > 0 and for any bounded interval I ⊂ R + 0 . This estimate and condition (3.22) imply the convergence in (3.21).

( 1 +Theorem 3 . 7 .F( 1 +

 1371 for τ > 0 and for any bounded interval I of R + 0 . Applying (3.21), we obtain limτ →+0 sup t∈I tS(τ )) -1 -(1 + tH) -1 ≤ ε,for any ε > 0. This completes the proof of(3.22). Now we are in position to prove another version of Theorem 3.3 for the operator-norm Chernoff approximation formula. We relax the restriction I ⊂ R + to condition I ⊂ R + 0 , but for (3.8) instead of (3.7). Let F (•) : R + 0 -→ L(H) be a measurable family of nonnegative self-adjoint contractions such that F (0) = 1. Let self-adjoint family {S(τ )} τ >0 be defined by(2.1)-(2.2), where H is non-negative self-adjoint operator in H. Then (3.23) lim η→∞ sup t∈I (t/η) η -e -tH = 0, for any bounded interval I ⊂ R + 0 if and only if (3.24) lim η→∞ sup t∈I tS(t/η)) -1 -(1 + tH) -1 = 0, is satisfied for any bounded interval I ⊂ R + 0 . Proof. By (3.14) and by assumption (3.24), we obtain the limit (3.23). Conversely, using (3.15) and assumption (3.23), one gets (3.21) for τ = t/η and for any bounded interval I ⊂ R + 0 . Then application of Theorem 3.6 yields (3.24).

(4.9) 1 0 1 0

 11 dy e -y e -ytS(τ ) -e -ytH = 1 τ /t dy e -y e -ytS(τ ) -e -ytH + τ /t 0 dy e -y e -ytS(τ ) -e -ytH . Hence, by (4.8) we obtain e -ytS(τ ) -e -ytH ≤ (1 + c ρ ) τ ty ρ , for τ, t, y ∈ (0, 1] and τ /t ≤ y. This yields the estimate (4.10) 1 τ /t dy e -y e -ytS(τ ) -e -ytH ≤ (1 + c ρ ) dy e -y y -ρ τ t ρ , for 0 < τ ≤ t ≤ 1 and ρ ∈ (0, 1). Since ρ < 1, one gets (4.11) τ /t 0 dy e -y e -ytS(τ ) -e -ytH ≤ 2 τ t ρ .

F

  (t/η) η -e -tH ≤ C I ρ 1 η ρ , holds for η ≥ 1. Proof. Let N ∈ N such that I ⊆ [0, N ]. Then representation

:

  = c ρ N 1+ρ , we get the proof of the theorem for I = [0, N ]. Since for any bounded interval I one can always find a N ∈ N such that I ⊆ [0, N ], this completes the proof.To extend this result to I = R + 0 one needs global conditions for 0 < τ ≤ t < ∞.

Theorem 4 . 3 .

 43 Let F (•) : R + 0 -→ L(H) be a measurable family of nonnegative self-adjoint contractions such that F (0) = 1. Let self-adjoint family {S(τ )} τ >0 be defined by (2.1)-(2.2), where H is a non-negative self-adjoint operator in H.

( 1 +

 1 tS(τ )) -1 -(1 + tH) -1 ≤ M R +

( 4 . 1 1 η , where c R + 1 : 1 η5. Concluding remarks 1 .

 41111 24)F (t/η) η -e -tH ≤ c R + = 1 + c M R + 1 for interval t ∈ [0, εη).Now let t ≥ εη. Then by assumption (4.20) we have (4.25)F (t/η) η ≤ (1 -δ ε ) η = e η ln(1-δε) , t ≥ ηε.Note that H ≥ µ1 implies e -tH ≤ e -ηεµ for t ≥ ηε. This together with (4.24) and (4.25) yield the estimateF (t/η) η -e -tH ≤ c R + 1 + e η ln(1-δε) + e -ηεµ ,for ε > 0, cf. (4.23) and for any t ≥ 0. Since c 1 := sup η≥1 η(e η ln(1-δε) +e -ηεµ ) < ∞, there exists constant c R + 1 := c R + 1 + c 1 such that (4.18) is valid for η ≥ 1 and infinite interval I = R + 0 Let the Kato functions f, g ∈ K 2 (Definition 1.1) and a measurable family of non-negative self-adjoint contractions with F (0) = 1 be defined byF (t) := g(tB) 1/2 f (tA)g(tB) 1/2 , t ≥ 0.Here A and B are positive self-adjoint operators in a Hilbert space H with domains dom A and dom B such that the operator sumC := A + B is self-adjoint on dom C = dom A ∩ dom B.Then by (1.2) the family {F (t)} t≥0 satisfies condition (4.20), i.e., Theorem 4.5 yields (1.5) in Proposition 1.2 for H = C and for discrete choice of continuous parameter: η = n, where n ∈ N.

Acknowledgements. I thank the referee for very useful remarks and suggestions.

Note that by Theorem 4.5, (4.24), one gets for c R + 1 > 1 and η = n -1 ≥ 1,

(5.1)

On the other hand, since f, g ∈ K 2 and C = A + B, one obtains estimates:

(5.2)

where γ[f ] := sup x>0 (1-f (x))/x and similarly for g. Collecting inequalities (5.1) and ( 5.2), we get for some Γ > 0 the estimate

that proves in (1.6) the asymptotic for n → ∞.

3. The proof of optimality of the asymptotic (1.5) and (1.6) is a subtle matter, 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]. To this aim, one has to establish for convergence an estimate from below and also an example, where the operator-norm convergence is broken if operator A + B is not self-adjoint, but only essentially self-adjoint.

In the present paper we developed the lifting topology of convergence for self-adjoint Chernoff approximation. It yields optimal estimate for the rate of convergence for Trotter-Kato product formulae. For non-self-adjoint case one uses other schemes essentially based on analyticity of semigroups, see [START_REF] Cachia | Operator-norm convergence of the Trotter product formula for holomorphic semigroups[END_REF][START_REF] Cachia | Operator-norm approximation of semigroups by quasisectorial contractions[END_REF]. The results for quasi-sectorial contractions [START_REF] Zagrebnov | Quasi-sectorial contractions[END_REF] improved by the 1/ 3 √ n -Theorem [START_REF] Zagrebnov | Comments on the Chernoff √ n-lemma[END_REF], are still not sufficiently refined to yield optimality for estimates of the rate of convergence.