Norm Estimates for Selfadjoint Toeplitz Operators on the Fock Space

An estimate for the norm of selfadjoint Toeplitz operators with a radial, bounded and integrable symbol is obtained. This emphasizes the fact that the norm of such operator is strictly less than the supremum norm of the symbol. Consequences for time-frequency localization operators are also given.


Introduction
The Bargmann-Fock space F 2 (C) is the Hilbert space consisting of those analytic functions f ∈ H (C) such that The normalized monomials e n (z) = π n n! 1 2 z n , n ≥ 0, form an orthonormal basis. For a fixed a ∈ C the translation operator is an isometry (see [14,Proposition 2.38]). We denote dλ(z) = e −π |z| 2 d A(z), so F 2 (C) is a closed subspace of L 2 (C, dλ). The orthogonal projection is the integral operator For a measurable and bounded function F on C the Toeplitz operator with symbol F is defined as The systematic study of Toeplitz operators on the Fock space started in [3,4]. Since then it has been a very active research area. We refer to [14,Chapter 6], where boundedness and membership in the Schatten classes is discussed. It is obvious that is a bounded operator and In particular, T F ≤ 1 whenever F ∞ ≤ 1. If moreover T F is compact, which happens for instance when F ∈ L 1 (C), then T F is strictly less than 1 but, as far as we know, no precise estimate for the norm is known. The main result of the paper gives a bound for T F in the case that the symbol F is radial, real-valued, and satisfies some integrability condition. For Toeplitz operators with radial symbols we refer to [11]. Besides Toeplitz operators on the Fock space we consider time-frequency localization operators with Gaussian window, also known as anti-Wick operators. They were introduced by Daubechies [7] as filters in signal analysis and can be obtained from Toeplitz operators on the Fock space after applying Bargmann transform.

Toeplitz Operators on the Fock Space
The Toeplitz operator defined by a real valued symbol F is self-adjoint. This is immediate from the identity for all f , g ∈ F 2 (C). In this case we have A symbol F is said to be radial with respect to a ∈ C if F(z) = g(|z − a|) for some bounded and measurable function g on [0, +∞). The main result of the paper is as follows.
be a real-valued and radial symbol with respect to a ∈ C. Then An expression for the norm of Toeplitz operators with radial symbols can be found in [11] but it is unclear how the estimate provided by Theorem 1 can be obtained from it.
For the proof we will need some auxiliary results. First we observe that for |F(z)| = g(|z|) and f = ∞ n=0 b n e n we have, after changing to polar coordinates, For the last identity observe that (b) For a general measurable set I with finite measure the conclusion follows from part (a) and the fact that for every ε > 0 there is a set J , finite union of bounded intervals, with the property that |J \ I | + |I \ J | ≤ ε.
Proof We denote by n the number of indexes k such that 0 < ε k < 1 and we proceed by induction on n. For n = 0 this is the content of Lemma 1. Let us now assume n = 1. Let 1 ≤ j ≤ N be the coordinate with the property that 0 < ε j < and check that for every 0 ≤ ε ≤ 1. In fact, ψ(0) ≤ 1 and ψ(1) ≤ 1 follow from Lemma 1. Moreover, the critical point ε 0 of ψ satisfies Let us assume that the Lemma holds for n = (0 ≤ < N ) and let n = + 1. We consider the function ψ : where ε 0 is a critical point of ψ. Proceeding as before,

Proof of Theorem 1
We first assume a = 0, that is, F is radial. After replacing F by Let us first assume where We used J k = t : t π ∈ I k and |J k | = 2π I k rdr. Theorem 1 is proved for g as in (1). Let us now assume that g ∞ ≤ 1 and g ∈ L 1 (R + , rdr) ∩ L ∞ (R + ). Then there is a sequence (g n ) n of step functions as in (1) We put F n (z) := g n (|z|). According to [12,Theorem 3.5] there is a constant K > 0 such that for every bounded symbol G, which implies We finally conclude In the case a = 0, the identity and the fact that W −a is an isometry gives the conclusion. We can also argue from the In particular, if Ω ⊂ C presents radial symmetry with respect to some point then for every f ∈ F 2 (C). The question arises whether inequality (3) holds for every subset Ω. This is related to a conjecture by Abreu and Speckbacher in [1] (see the next section). We do not have an answer to this question except for monomials or its translates.

Example 1
Let k w = e − π 2 |w| 2 K w be the normalized reproducing kernel of F 2 (C). Then, for every set Ω ⊂ C with finite measure we have and the conclusion follows from the fact that the last integral attains its maximum when Ω is a disc centered at w (see the comment after [1, Conjecture 1]).
It is easy to check that when Ω is a disc centered at point ω the inequality in Example 1 is an identity.

Proposition 1
Let Ω ⊂ R 2 be a set with finite measure. Then, for every n ∈ N and a ∈ C, we can assume that a = 0. For every θ ∈ [0, 2π ] we denote where I θ = t = πr 2 : r ∈ Ω θ .
Since |Ω| < ∞ then a.e. θ ∈ [0, 2π ] we have Moreover, by Lemma 1, Finally we consider the convex function f (t) = e −t − 1 and the probability measure dθ 2π and put h(θ ) = |I θ |. Jensen's inequality gives We finish the section with some examples of sets Ω with infinite Lebesgue measure for which the Toeplitz operator with symbol F = χ Ω has norm as small as we want.

Proposition 2 For every ε > 0 there exists Ω with infinite Lebesgue measure such that
Proof Let K > 0 as in (2) and let (Ω n ) n be a sequence of bounded sets with Lebesgue measure |Ω n | = ε K and such that dist(Ω n , Ω m ) > 2 whenever n = m, and take Ω = ∪ n∈N Ω n . Since each disc D(z, 1) meets at most one set Ω n we have |Ω∩D(z, 1)| ≤ ε K . The estimate (2) turns out T χ Ω ≤ ε, which gives the conclusion.

Time-Frequency Localization Operators
For F ∈ L 1 (C) we denote by H F : L 2 (R) → L 2 (R) the localization operator Here h 0 (t) = 2 1/4 e −π t 2 is the Gaussian and π(z) is the time-frequency shift, defined for z = x + iω as In case F is the characteristic function of a set Ω we write H Ω instead of H χ Ω . We refer to [5] or [6,Chapter 4] for general facts concerning localization operators.
For f , g ∈ L 2 (R), the expression is the short time Fourier transform of f with window g, known as Gabor transform in the case where the window g = h 0 is the Gaussian.
If F is real-valued then H F is a selfadjoint operator on L 2 (R), hence There is a connection between localization operators and Toeplitz operators on the Fock space via the Bargmann transform. The Bargmann transform is the surjective and unitary operator defined as It was introduced in [2] and has the important property that the Hermite functions are mapped into normalized analytic monomials. More precisely, B(h n ) = e n , where h n is defined via the so called Rodrigues formula as Then (h n ) n≥0 forms an orthonormal basis for L 2 (R). The Gabor transform of Hermite functions is well-known (see for instance [9, Chapter 1.9]). In fact, for z = x + iξ, h n , π(z)h 0 = e −iπ xξ − π 2 |z| 2 π n n! z n .
Since for z = x + iξ we have ( [10, 3.4.1]) then, for every f ∈ L 2 (R) and F ∈ L 1 (C) ∩ L ∞ (C) we obtain Consequently, all the estimates in the previous section can be translated into estimates concerning localization operators. Abreu, Speckbacher conjecture in [1] that, among all the sets with a given measure, H Ω attains its maximum when Ω is a disc, up to perturbations of Lebesgue measure zero. This turns out to be equivalent to the validity of inequality (3) for every function in the Fock space or, equivalently, to the fact that In this regard it is worth noting that Nazarov [13] proved the existence of two absolute constants A, B such that for every f ∈ L 2 (R) and for any pair (S, ) of sets with finite measure. Also, it follows from [8,Theorem 4.1] that for every set Ω ⊂ R 2 thin at infinity and for every g ∈ L 2 (R) there exist a constant C > 0 such that From Theorem 1 and Proposition 1 we get the following.
be a real-valued and radial symbol with respect to a ∈ C. Then

Corollary 2
Let Ω ⊂ R 2 be a set with finite measure. Then, for every n ∈ N, We fix a non-zero window g ∈ L 2 (R). The modulation space M 1 (R), also known as Feichtinger algebra, is the set of tempered distributions f ∈ S (R) such that The use of different windows g in the definition of M 1 (R) yields the same spaces with equivalent norms. It is well known that M 1 (R) is continuously included in L 2 (R) and whenever f ∈ M 1 (R) and g 2 = 1. See for instance [10, 3.2.1] for the first identity.

Proposition 3
Let Ω ⊂ R 2 be a set with finite measure. Then, for every f ∈ M 1 (R) and n ∈ N 0 we have Proof It suffices to prove the proposition under the additional assumption that V h n f 1 = 1. Fixed n ∈ N 0 we consider the set Then We have for every g ∈ B • , which means that f ∈ B •• . According to the bipolar theorem, where each f k is in the absolutely convex hull of B. For each k ∈ N we can find scalars (α j ) N j=1 and points (z j ) N j=1 such that f k = N j=1 α j π(z j )h n and N j=1 |α j | ≤ 1. Then Finally, The next result is a direct consequence of Proposition 2.
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Conflict of interest
The authors declare that they have no conflict of interest.
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