Norm Estimates for Selfadjoint Toeplitz Operators on the Fock Space

Abstract

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.

1 Introduction

The Bargmann-Fock space ${{\mathcal {F}}}^2({\mathbb {C}})$ is the Hilbert space consisting of those analytic functions $f\in H({\mathbb {C}})$ such that

$$\begin{aligned} \Vert f\Vert _{{{\mathcal {F}}}}^2 = \int _{{\mathbb {C}}}|f(z)|^2 e^{-\pi |z|^2}\ dA(z) < +\infty , \end{aligned}$$

where dA(z) denotes the Lebesgue measure. ${{\mathcal {F}}}^2({\mathbb {C}})$ admits a reproducing kernel $K_w(z) = e^{\pi {\overline{w}}z},$ which means that

$$\begin{aligned}f(w) = \langle f, K_w\rangle ,\ \ f\in {{\mathcal {F}}}^2({\mathbb {C}}). \end{aligned}$$

The normalized monomials

$$\begin{aligned} e_n(z) = \left( \frac{\pi ^n}{n!}\right) ^{\frac{1}{2}} z^n,\ \ n\ge 0, \end{aligned}$$

form an orthonormal basis. For a fixed $a\in {\mathbb {C}}$ the translation operator

$$\begin{aligned} W_a:{{\mathcal {F}}}^2({\mathbb {C}})\rightarrow {{\mathcal {F}}}^2({\mathbb {C}}),\ \left( W_a f\right) (z) = f(z-a)e^{-\frac{\pi }{2} |a|^2 + \pi z{\overline{a}}}, \end{aligned}$$

is an isometry (see [14, Proposition 2.38]). We denote $d\lambda (z) = e^{-\pi |z|^2} dA(z),$ so ${{\mathcal {F}}}^2({\mathbb {C}})$ is a closed subspace of $L^2({\mathbb {C}}, d\lambda ).$ The orthogonal projection

$$\begin{aligned} P:L^2({\mathbb {C}}, d\lambda )\rightarrow {{\mathcal {F}}}^2({\mathbb {C}}) \end{aligned}$$

is the integral operator

$$\begin{aligned} \left( Pf\right) (z) = \int _{{\mathbb {C}}} f(w)K_w(z)\ d\lambda (w). \end{aligned}$$

For a measurable and bounded function F on ${\mathbb {C}}$ the Toeplitz operator with symbol F is defined as

$$\begin{aligned} T_F(f)(z) = P(Ff)(z) = \int _{{\mathbb {C}}} F(w)f(w)K_w(z)\ d\lambda (w). \end{aligned}$$

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

$$\begin{aligned}T_F:{{\mathcal {F}}}^2({\mathbb {C}})\rightarrow {{\mathcal {F}}}^2({\mathbb {C}}) \end{aligned}$$

is a bounded operator and

$$\begin{aligned} \Vert T_F(f)\Vert \le \Vert Ff\Vert _{L^2({\mathbb {C}}, d\lambda )} \le \Vert F\Vert _\infty \cdot \Vert f\Vert . \end{aligned}$$

In particular, $\Vert T_F\Vert \le 1$ whenever $\Vert F\Vert _\infty \le 1.$ If moreover $T_F$ is compact, which happens for instance when $F\in L^1({\mathbb {C}}),$ then $\Vert T_F\Vert$ 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 $\Vert T_F\Vert$ 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.

2 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

$$\begin{aligned} \langle T_F(f), g\rangle = \int _{{\mathbb {C}}} F(z) f(z)\overline{g(z)}d\lambda (z) \end{aligned}$$

for all $f,g\in {{\mathcal {F}}}^2({\mathbb {C}}).$ In this case we have

$$\begin{aligned} \Vert T_F\Vert = \sup _{\Vert f\Vert =1}\left| \langle T_F(f), f\rangle \right| \le \sup _{\Vert f\Vert =1}\int _{{\mathbb {C}}} |F(z)|\cdot |f(z)|^2d\lambda (z). \end{aligned}$$

A symbol F is said to be radial with respect to $a\in {\mathbb {C}}$ if $F(z) = g(|z-a|)$ for some bounded and measurable function g on $[0, +\infty ).$ The main result of the paper is as follows.

Theorem 1

Let $F\in L^1({\mathbb {C}})\cap L^\infty ({\mathbb {C}})$ be a real-valued and radial symbol with respect to $a\in {\mathbb {C}}.$ Then

$$\begin{aligned} \Vert T_F\Vert \le \Vert F\Vert _\infty \Bigg (1 - \exp \Bigg (-\frac{\Vert F\Vert _1}{\Vert F\Vert _\infty }\Bigg )\Bigg ).\end{aligned}$$

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 $f = \sum _{n=0}^\infty b_n e_n$ we have, after changing to polar coordinates,

$$\begin{aligned} \int _{{\mathbb {C}}} |F(z)|\cdot |f(z)|^2d\lambda (z)= & {} \sum _{n=0}^\infty |b_n|^2\int _{{\mathbb {C}}}g(|z|)|e_n(z)|^2\ d\lambda (z) \\= & {} \sum _{n=0}^\infty |b_n|^2 2\pi \int _0^\infty g(r)\pi ^n\frac{r^{2n+1}}{n!}e^{-\pi r^2}\ dr \\= & {} \sum _{n=0}^\infty |b_n|^2 \int _0^\infty g\Big (\sqrt{\frac{t}{\pi }}\Big )\frac{t^n}{n!} e^{-t}\ dt. \end{aligned}$$

The d-dimensional Lebesgue measure of a set $\varOmega \subset {\mathbb {R}}^d$ is denoted $|\varOmega |$ both for $d=1$ and $d=2.$

Lemma 1

Let $I\subset [0, +\infty )$ be a measurable set with finite Lebesgue measure. Then

$$\begin{aligned} \frac{1}{n!}\int _I s^ne^{-s}\ ds \le 1-e^{-|I|}. \end{aligned}$$

Proof

• (a) We first assume that I is a finite union of bounded intervals. The function $h(s) = \frac{s^n}{n!}e^{-s}$ attains its absolute maximum at $s = n.$ Then h increases on [0, n] and decreases on $[n, +\infty ).$ We consider $a\le n\le b$ such that

$$\begin{aligned} n - a = |I\cap [0, n]|\ ,\ b-n = |I\cap [n, +\infty )|.\end{aligned}$$

Then

$$\begin{aligned} \frac{1}{n!}\int _I s^n e^{-s}\ ds\le & {} \int _a^b h(s)\ ds = \frac{e^{-a}}{n!}\int _0^{b-a}(t+a)^n e^{-t}\ dt \\= & {} \sum _{k=0}^n \left( {\begin{array}{c}n\\ k\end{array}}\right) \frac{a^{n-k}}{n!}e^{-a}\int _0^{|I|} t^k e^{-t}\ dt \\= & {} \sum _{k=0}^n \frac{a^{n -k}}{(n -k)!}e^{-a}\frac{1}{k!}\int _0^{|I|} t^k e^{-t}\ dt\\\le & {} \sup _{0\le k\le n}\frac{1}{k!}\int _0^{|I|} t^k e^{-t}\ dt = \int _0^{|I|}e^{-t}\ dt. \end{aligned}$$

For the last identity observe that

$$\begin{aligned} \frac{1}{k!}\int _0^{s} t^k e^{-t}\ dt = 1 - e^{-s}\sum _{j=0}^k \frac{s^j}{j!}. \end{aligned}$$

• (b) For a general measurable set I with finite measure the conclusion follows from part (a) and the fact that for every $\varepsilon > 0$ there is a set J,  finite union of bounded intervals, with the property that

$$\begin{aligned} |J\setminus I| + |I\setminus J| \le \varepsilon . \end{aligned}$$

$\square$

Lemma 2

Let $(I_k)_{k=1}^N$ be disjoint sets with finite measure and $0\le \varepsilon _k\le 1$ for every $1\le k\le N.$ Then, for every $p\in {\mathbb {N}}_0$ we have

$$\begin{aligned} \sum _{k=1}^N \varepsilon _k\int _{I_k}\frac{t^p}{p!}e^{-t}\ dt \le 1 - \exp \left( -\sum _{k=1}^N \varepsilon _k |I_k|\right) . \end{aligned}$$

Proof

We denote by n the number of indexes k such that $0< \varepsilon _k < 1$ and we proceed by induction on n. For $n = 0$ this is the content of Lemma 1. Let us now assume $n = 1.$ Let $1\le j\le N$ be the coordinate with the property that $0< \varepsilon _j <$ and check that

$$\begin{aligned} \psi (\varepsilon ):= \sum _{k\ne j}\int _{I_k}\frac{t^p}{p!}e^{-t}\ dt + \varepsilon \int _{I_j}\frac{t^p}{p!}e^{-t}\ dt + \exp \left( -\sum _{k\ne j}|I_k| - \varepsilon |I_j|\right) \le 1\end{aligned}$$

for every $0 \le \varepsilon \le 1.$ In fact, $\psi (0) \le 1$ and $\psi (1)\le 1$ follow from Lemma 1. Moreover, the critical point $\varepsilon _0$ of $\psi$ satisfies

$$\begin{aligned} \int _{I_j}\frac{t^p}{p!}e^{-t}\ dt = |I_j|\exp \left( -\sum _{k\ne j}|I_k| - \varepsilon _0 |I_j|\right) . \end{aligned}$$

Hence

$$\begin{aligned} \psi (\varepsilon _0)= & {} \sum _{k\ne j}\int _{I_k}\frac{t^p}{p!}e^{-t}\ dt + \varepsilon _0|I_j|\exp \left( -\sum _{k\ne j}|I_k| - \varepsilon _0 |I_j|\right) \\&+ \exp \left( -\sum _{k\ne j}|I_k| - \varepsilon |I_j|\right) \\= & {} \sum _{k\ne j}\int _{I_k}\frac{t^p}{p!}e^{-t}\ dt + \left( 1 + \varepsilon _0|I_j|\right) \exp \left( -\sum _{k\ne j}|I_k| - \varepsilon |I_j|\right) . \end{aligned}$$

Since

$$\begin{aligned} 1 + \varepsilon _0|I_j| \le \exp \big (\varepsilon _0 |I_j|\big ) \end{aligned}$$

we conclude

$$\begin{aligned} \psi (\varepsilon _0) \le \sum _{k\ne j}\int _{I_k}\frac{t^p}{p!}e^{-t}\ dt + \exp \left( -\sum _{k\ne j}|I_k|\right) \le 1. \end{aligned}$$

Let us assume that the Lemma holds for $n = \ell$ ($0\le \ell < N$) and let $n = \ell +1.$ We consider the function $\psi :[0,1]^{\ell +1}\rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} \psi ({\varvec{\varepsilon }}):= \sum _{k=1}^{\ell +1}\varepsilon _k \int _{I_k}\frac{t^p}{p!}e^{-t}\ dt + \sum _j\int _{J_j}\frac{t^p}{p!}e^{-t}\ dt + \exp \left( -\sum _k\varepsilon _k |I_k| - \sum _j |J_j|\right) \end{aligned}$$

for ${\varvec{\varepsilon }} = (\varepsilon _1, \ldots , \varepsilon _{\ell +1}).$ The induction hypothesis means that $\psi ({\varvec{\varepsilon }}) \le 1$ whenever ${\varvec{\varepsilon }}$ is in the boundary of $[0,1]^{\ell +1}.$ The lemma is proved after checking that $\psi ({\varvec{\varepsilon }}_0) \le 1,$ where ${\varvec{\varepsilon }_0}$ is a critical point of $\psi .$ Proceeding as before,

$$\begin{aligned} \psi ({\varvec{\varepsilon }}_0)= & {} \left( \sum _{k=1}^{\ell +1}\varepsilon _k |I_k| + 1\right) e^{-\sum _k \varepsilon _k |I_k|} e^{-\sum _j |J_j|} + \sum _j \int _{J_j}\frac{t^p}{p!}e^{-t}\ dt \\\le & {} \exp \left( -\sum _j |J_j|\right) + \sum _j \int _{J_j}\frac{t^p}{p!}e^{-t}\ dt \le 1. \end{aligned}$$

$\square$

Proof of Theorem 1

We first assume $a = 0,$ that is, F is radial. After replacing F by $G = \frac{F}{\Vert F\Vert _\infty }$ if necessary we can assume that $\Vert F\Vert _\infty = 1.$ Since F is radial we have $F(z) = g\big (|z|\big ).$ We aim to prove that

$$\begin{aligned} \int _{{\mathbb {C}}}\left| g\big (|z|\big )\right| \cdot \left| f(z)\right| ^2 e^{-\pi |z|^2}\ dA(z) \le 1 - \exp \left( -2\pi \int _0^\infty r\left| g(r)\right| \ dr\right) \end{aligned}$$

for every entire function $f(z) = \sum _{p=0}^\infty b_p e_p$ such that $\sum _{p=0}^\infty |b_p|^2 = 1.$ We have

$$\begin{aligned} \int _{{\mathbb {C}}}\left| g\big (|z|\big )\right| \cdot \left| F(z)\right| ^2 e^{-\pi |z|^2}\ dA(z) = \sum _{p=0}^\infty |b_p|^2\int _0^\infty \left| g\left( \sqrt{\frac{t}{\pi }}\right) \right| \cdot \frac{t^p}{p!}e^{-t}\ dt. \end{aligned}$$

Let us first assume

$$\begin{aligned} g = \sum _{k=1}^N \varepsilon _k \chi _{I_k},\ \left| \varepsilon _k\right| \le 1, \end{aligned}$$

(1)

where $(I_k)_{k=1}^N$ are disjoint intervals. Then, Lemma 1 gives

$$\begin{aligned} \sum _{p=0}^\infty |b_p|^2\int _0^\infty \left| g\left( \sqrt{\frac{t}{\pi }}\right) \right| \cdot \frac{t^p}{p!}e^{-t}\ dt\le & {} 1 - \exp \left( -\sum _{k=1}^N |\varepsilon _k| |J_k|\right) \\= & {} 1 - \exp \left( -2\pi \int _0^\infty r|g(r)|\ dr\right) \\= & {} 1 - \exp \left( -\Vert F\Vert _1\right) . \end{aligned}$$

We used $J_k = \left\{ t:\ \sqrt{\frac{t}{\pi }}\in I_k\right\}$ and $|J_k| = 2\pi \int _{I_k}r dr.$ Theorem 1 is proved for g as in (1). Let us now assume that $\Vert g\Vert _\infty \le 1$ and $g\in L^1({\mathbb {R}}^+,rdr)\cap L^\infty ({\mathbb {R}}^+).$ Then there is a sequence $(g_n)_n$ of step functions as in (1) such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _0^\infty |g_n(r)-g(r)|\ r dr = 0. \end{aligned}$$

We put $F_n(z):= g_n(|z|).$ According to [12, Theorem 3.5] there is a constant $K > 0$ such that

$$\begin{aligned} \Vert T_G\Vert \le K\sup _{z\in {{\mathbb {C}}}}\int _{D(z,1)}|G|\ dA \end{aligned}$$

(2)

for every bounded symbol G,  which implies

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert T_F - T_{F_n}\Vert \le K\lim _{n\rightarrow \infty }\Vert F_n - F\Vert _1 = 0. \end{aligned}$$

We finally conclude

$$\begin{aligned} \Vert T_F\Vert \le 1 - \exp \big (-\Vert F\Vert _1\big ). \end{aligned}$$

In the case $a\ne 0,$ the identity

$$\begin{aligned} \int _{{\mathbb {C}}} g\big (|z-a|\big )|f(z)|^2 d\lambda (z) = \int _{{\mathbb {C}}} g\big (|u|\big )\left| (W_{-a}f)(u)\right| ^2 d\lambda (u) \end{aligned}$$

and the fact that $W_{-a}$ is an isometry gives the conclusion. We can also argue from the fact that $W_{-a}\circ T_F = T_G\circ W_{-a},$ where $G(z) = g\big (|z|\big ).$ $\square$

In particular, if $\varOmega \subset {{\mathbb {C}}}$ presents radial symmetry with respect to some point then

$$\begin{aligned} \int _\varOmega \left| f(z)\right| ^2 d\lambda (z) \le \Big (1 - e^{-|\varOmega |}\Big )\cdot \int _{{{\mathbb {C}}}}\left| f(z)\right| ^2 d\lambda (z) \end{aligned}$$

(3)

for every $f\in {{\mathcal {F}}}^2({{\mathbb {C}}}).$

The question arises whether inequality (3) holds for every subset $\varOmega .$ 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 $k_w = e^{-\frac{\pi }{2}|w|^2}K_w$ be the normalized reproducing kernel of ${{\mathcal {F}}}^2({{\mathbb {C}}}).$ Then, for every set $\varOmega \subset {{\mathbb {C}}}$ with finite measure we have

$$\begin{aligned} \int _\varOmega \left| k_w(z)\right| ^2 d\lambda (z) \le 1 - e^{-|\varOmega |}. \end{aligned}$$

Proof

In fact, $k_w = W_w\left( e_0\right) .$ Hence

$$\begin{aligned} \int _\varOmega |k_w(z)|^2 d\lambda (z) = \int _{\varOmega -w} d\lambda (z) \end{aligned}$$

and the conclusion follows from the fact that the last integral attains its maximum when $\varOmega$ is a disc centered at w (see the comment after [1, Conjecture 1]). $\square$

It is easy to check that when $\varOmega$ is a disc centered at point $\omega$ the inequality in Example 1 is an identity.

Proposition 1

Let $\varOmega \subset {\mathbb {R}}^2$ be a set with finite measure. Then, for every $n\in {\mathbb {N}}$ and $a\in {{\mathbb {C}}},$

$$\begin{aligned} \int _\varOmega \left| W_a(e_n)(z)\right| ^2 d\lambda (z)\le 1-e^{-|\varOmega |}. \end{aligned}$$

Proof

Since

$$\begin{aligned} \int _\varOmega \left| W_a(e_n)(z)\right| ^2 d\lambda (z) = \int _{\varOmega -a} \left| e_n(z)\right| ^2 d\lambda (z) \end{aligned}$$

we can assume that $a = 0.$ For every $\theta \in [0,2\pi ]$ we denote

$$\begin{aligned} \varOmega _\theta = \left\{ r \ge 0:\ re^{i\theta }\in \varOmega \right\} . \end{aligned}$$

Then

$$\begin{aligned} \int _\varOmega \left| e_n(z)\right| ^2 d\lambda (z)= & {} \frac{\pi ^n}{n!}\int _\varOmega \left| z^n\right| ^2 e^{-\pi |z|^2}\ dA(z)\\= & {} \frac{\pi ^n}{n!}\int _0^{2\pi }\Bigg (\int _{\varOmega _\theta }r^{2n}e^{-\pi r^2}2\pi r\ dr\Bigg ) \frac{d\theta }{2\pi }\\= & {} \int _0^{2\pi }\Bigg (\int _{I_\theta }\frac{t^n}{n!} e^{-t}\ dt\Bigg ) \frac{d\theta }{2\pi }, \end{aligned}$$

where

$$\begin{aligned} I_\theta = \left\{ t = \pi r^2:\ r\in \varOmega _\theta \right\} . \end{aligned}$$

Since $|\varOmega | < \infty$ then a.e. $\theta \in [0,2\pi ]$ we have

$$\begin{aligned} |I_\theta | = 2\pi \int _{\varOmega _\theta }r dr < +\infty . \end{aligned}$$

Moreover, by Lemma 1,

$$\begin{aligned} \int _0^{2\pi }\left( \int _{I_\theta }\frac{t^n}{n!} e^{-t}\ dt\right) \frac{d\theta }{2\pi } \le \int _0^{2\pi } \left( 1-e^{-|I_\theta |}\right) \frac{d\theta }{2\pi }. \end{aligned}$$

Finally we consider the convex function $f(t) = e^{-t}-1$ and the probability measure $\frac{d\theta }{2\pi }$ and put $h(\theta ) = |I_\theta |.$ Jensen’s inequality gives

$$\begin{aligned} f\left( \int _0^{2\pi }h(\theta )\frac{d\theta }{2\pi }\right) \le \int _0^{2\pi }f\left( h(\theta )\right) \frac{d\theta }{2\pi }, \end{aligned}$$

which means

$$\begin{aligned} \int _0^{2\pi } \left( 1-e^{-|I_\theta |}\right) \frac{d\theta }{2\pi }\le & {} 1 - \exp \left( -\int _0^{2\pi }|I_\theta |\ \frac{d\theta }{2\pi }\right) \\= & {} 1 - \exp \left( -\int _0^{2\pi }\left( \int _{\varOmega _\theta } r\ dr\right) \ d\theta \right) \\= & {} 1 - e^{-|\varOmega |}. \end{aligned}$$

$\square$

We finish the section with some examples of sets $\varOmega$ with infinite Lebesgue measure for which the Toeplitz operator with symbol $F=\chi _\varOmega$ has norm as small as we want.

Proposition 2

For every $\varepsilon > 0$ there exists $\varOmega$ with infinite Lebesgue measure such that

$$\begin{aligned} \int _\varOmega |f(z)|^2 d\lambda (z) \le \varepsilon \int _{{{\mathbb {C}}}} |f(z)|^2 d\lambda (z) \end{aligned}$$

for every $f\in {{\mathcal {F}}}^2.$

Proof

Let $K > 0$ as in (2) and let $(\varOmega _n)_n$ be a sequence of bounded sets with Lebesgue measure $|\varOmega _n| = \frac{\varepsilon }{K}$ and such that $\text{ dist } (\varOmega _n, \varOmega _m) > 2$ whenever $n\ne m,$ and take $\varOmega = \cup _{n\in {{\mathbb {N}}}} \varOmega _n.$ Since each disc D(z, 1) meets at most one set $\varOmega _n$ we have $|\varOmega \cap D(z,1)|\le \frac{\varepsilon }{K}.$ The estimate (2) turns out $\Vert T_{\chi _\varOmega }\Vert \le \varepsilon ,$ which gives the conclusion. $\square$

3 Time-Frequency Localization Operators

For $F\in L^1({{\mathbb {C}}})$ we denote by $H_F:L^2({\mathbb {R}})\rightarrow L^2({\mathbb {R}})$ the localization operator

$$\begin{aligned} H_F f = \int _{{\mathbb {C}}} F(z)\ \langle f, \pi (z)h_0\rangle \ \pi (z)h_0\ dA(z). \end{aligned}$$

Here $h_0(t) = 2^{1/4} e^{-\pi t^2}$ is the Gaussian and $\pi (z)$ is the time-frequency shift, defined for $z = x+i\omega$ as

$$\begin{aligned} \big (\pi (z)f\big )(t) = e^{2\pi i \omega t}f(t-x),\ f\in L^2({\mathbb {R}}). \end{aligned}$$

In case F is the characteristic function of a set $\varOmega$ we write $H_\varOmega$ instead of $H_{\chi _\varOmega }.$ We refer to [5] or [6, Chapter 4] for general facts concerning localization operators.

For $f,g\in L^2({\mathbb {R}}),$ the expression

$$\begin{aligned} \big (V_g f\big )(z) := \langle f, \pi (z)g\rangle \end{aligned}$$

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 $L^2({\mathbb {R}}),$ hence

$$\begin{aligned} \Vert H_F\Vert = \sup _{\Vert f\Vert _2 = 1}\left| \langle H_F f,\ f\rangle \right| \le \sup _{\Vert f\Vert _2 = 1}\int _{{{\mathbb {C}}}} |F(z)|\cdot \left| \left( V_{h_0} f\right) (z)\right| ^2 dA(z). \end{aligned}$$

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

$$\begin{aligned} {{\mathcal {B}}}:L^2({\mathbb {R}}) \rightarrow {{\mathcal {F}}}^2({{\mathbb {C}}}) \end{aligned}$$

defined as

$$\begin{aligned} \big ({{\mathcal {B}}}f\big )(z) = 2^{1/4}\int _{{\mathbb {R}}}f(t) e^{2\pi t z -\pi t^2-\frac{\pi }{2}z^2}\ dt. \end{aligned}$$

It was introduced in [2] and has the important property that the Hermite functions are mapped into normalized analytic monomials. More precisely, ${{\mathcal {B}}}(h_n) = e_n,$ where $h_n$ is defined via the so called Rodrigues formula as

$$\begin{aligned} h_n(t) = \frac{2^{1/4}}{\sqrt{n!}}\left( \frac{-1}{2\sqrt{\pi }}\right) ^n e^{\pi t^2}\frac{d^n}{dt^n}\left( e^{-2\pi t^2}\right) ,\ \ n\ge 0. \end{aligned}$$

Then $\left( h_n\right) _{n\ge 0}$ forms an orthonormal basis for $L^2({\mathbb {R}}).$ The Gabor transform of Hermite functions is well-known (see for instance [9, Chapter 1.9]). In fact, for $z = x + i\xi ,$

$$\begin{aligned} \langle h_n, \pi (z)h_0\rangle = e^{-i\pi x\xi - \frac{\pi }{2}|z|^2}\sqrt{\frac{\pi ^n}{n!}}{\overline{z}}^n. \end{aligned}$$

(4)

Since for $z = x + i\xi$ we have ( [10, 3.4.1])

$$\begin{aligned} \big (V_{h_0} f\big )(x,-\xi ) = e^{i\pi x\xi }\cdot \big ({{\mathcal {B}}}f\big )(z)\cdot e^{-\frac{\pi |z|^2}{2}} \end{aligned}$$

then, for every $f\in L^2({\mathbb {R}})$ and $F\in L^1({{\mathbb {C}}})\cap L^\infty ({{\mathbb {C}}})$ we obtain

$$\begin{aligned} \int _{{{\mathbb {C}}}} |F(z)|\cdot \left| \left( V_{h_0} f\right) (z)\right| ^2 dA(z) = \int _{{{\mathbb {C}}}} |F(z)|\cdot \left| \big (Bf\big )(z)\right| ^2\ d\lambda (z). \end{aligned}$$

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, $\Vert H_\varOmega \Vert$ attains its maximum when $\varOmega$ 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

$$\begin{aligned} \Vert f\Vert _2^2 \le e^{|\varOmega |}\int _{{{\mathbb {C}}}\setminus \varOmega } \left| \big (V_{h_0}f\big )(z)\right| ^2 dA(z)\ \ \forall f\in L^2({\mathbb {R}}). \end{aligned}$$

In this regard it is worth noting that Nazarov [13] proved the existence of two absolute constants A, B such that

$$\begin{aligned} \Vert f\Vert _2^2 \le Ae^{B\cdot |S|\cdot |\Sigma |}\left( \int _{{\mathbb {R}}\setminus S} |f|^2 + \int _{{\mathbb {R}}\setminus \Sigma }|{\widehat{f}}|^2\right) \end{aligned}$$

for every $f\in L^2({\mathbb {R}})$ and for any pair $(S,\Sigma )$ of sets with finite measure. Also, it follows from [8, Theorem 4.1] that for every set $\varOmega \subset {{\mathbb {R}}}^{2}$ thin at infinity and for every $g\in L^2({{\mathbb {R}}})$ there exist a constant $C > 0$ such that

$$\begin{aligned} \Vert f\Vert _2^2 \le C\int _{{{\mathbb {C}}}\setminus \varOmega } \left| \big (V_{g}f\big )(z)\right| ^2 dA(z)\ \ \forall f\in L^2({\mathbb {R}}). \end{aligned}$$

From Theorem 1 and Proposition 1 we get the following.

Corollary 1

Let $F\in L^1({{\mathbb {C}}})\cap L^\infty ({{\mathbb {C}}})$ be a real-valued and radial symbol with respect to $a\in {{\mathbb {C}}}.$ Then

$$\begin{aligned} \Vert H_F\Vert \le \Vert F\Vert _\infty \Bigg (1 - \exp \Bigg (-\frac{\Vert F\Vert _1}{\Vert F\Vert _\infty }\Bigg )\Bigg ). \end{aligned}$$

Corollary 2

Let $\varOmega \subset {\mathbb {R}}^2$ be a set with finite measure. Then, for every $n\in {\mathbb {N}},$

$$\begin{aligned} \left| \langle H_\varOmega h_n,\ h_n\rangle \right| \le 1-e^{-|\varOmega |}. \end{aligned}$$

We fix a non-zero window $g\in L^2({\mathbb {R}}).$ The modulation space $M^1({\mathbb {R}}),$ also known as Feichtinger algebra, is the set of tempered distributions $f\in {{\mathcal {S}}}^\prime ({\mathbb {R}})$ such that

$$\begin{aligned} \Vert f\Vert _{M^1}:= \int _{{{\mathbb {C}}}}\left| \langle f, \pi (z)g\rangle \right| dA(z) < +\infty . \end{aligned}$$

The use of different windows g in the definition of $M^1({\mathbb {R}})$ yields the same spaces with equivalent norms. It is well known that $M^1({\mathbb {R}})$ is continuously included in $L^2({\mathbb {R}})$ and

$$\begin{aligned} \Vert f\Vert _2 = \Vert V_g f\Vert _ 2 \le \Vert V_g f\Vert _1 \end{aligned}$$

whenever $f\in M^1({\mathbb {R}})$ and $\Vert g\Vert _2 = 1.$ See for instance [10, 3.2.1] for the first identity.

Proposition 3

Let $\varOmega \subset {\mathbb {R}}^2$ be a set with finite measure. Then, for every $f\in M^1({\mathbb {R}})$ and $n\in {\mathbb {N}}_0$ we have

$$\begin{aligned} \int _\varOmega \left| \left( V_{h_0} f\right) (z)\right| ^2 dA(z) \le \Vert V_{h_n} f\Vert _1^2\cdot \big (1-e^{-|\varOmega |}\big ). \end{aligned}$$

Proof

It suffices to prove the proposition under the additional assumption that $\Vert V_{h_n}f\Vert _1 = 1.$ Fixed $n\in {\mathbb {N}}_0$ we consider the set

$$\begin{aligned} B:=\left\{ \pi (z)h_n:\ z\in {{\mathbb {C}}}\right\} \subset L^2({\mathbb {R}}). \end{aligned}$$

Then

$$\begin{aligned} B^\circ := \left\{ g\in L^2({\mathbb {R}}):\ \left| \langle g, \pi (z)h_n\rangle \right| \le 1\right\} = \left\{ g\in L^2({\mathbb {R}}):\ \Vert V_{h_n}g\Vert _\infty \le 1\right\} . \end{aligned}$$

We have

$$\begin{aligned} \left| \langle f, g\rangle \right| = \left| \langle V_{h_n}f, V_{h_n}g\rangle \right| \le \Vert V_{h_n}f\Vert _1\cdot \Vert V_{h_n}g\Vert _\infty \le 1 \end{aligned}$$

for every $g\in B^\circ ,$ which means that $f\in B^{\circ \circ }.$ According to the bipolar theorem,

$$\begin{aligned} f = L^2-\lim _{k\rightarrow \infty } f_k \end{aligned}$$

where each $f_k$ is in the absolutely convex hull of B. For each $k\in {\mathbb {N}}$ we can find scalars $(\alpha _j)_{j= 1}^N$ and points $(z_j)_{j=1}^N$ such that $f_k = \sum _{j=1}^N\alpha _j \pi (z_j)h_n$ and $\sum _{j=1}^N |\alpha _j| \le 1.$ Then

$$\begin{aligned} \Bigg (\int _\varOmega \left| \left( V_{h_0} f_k\right) (z)\right| ^2 dA(z)\Bigg )^{\frac{1}{2}}= & {} \Bigg (\int _\varOmega \left| \langle f_k, \pi (z)\varphi \rangle \right| ^2 dA(z)\Bigg )^{\frac{1}{2}} \\\le & {} \sum _{j=1}^N |\alpha _j|\Bigg (\int _\varOmega \left| \langle \pi (z_j)h_n, \pi (z)\varphi \rangle \right| ^2 dA(z)\Bigg )^{\frac{1}{2}}\\= & {} \sum _{j=1}^N |\alpha _j|\Bigg (\int _\varOmega \left| \langle h_n, \pi (z-z_j)\varphi \rangle \right| ^2 dA(z)\Bigg )^{\frac{1}{2}}\\= & {} \sum _{j=1}^N |\alpha _j|\Bigg (\int _{\varOmega -z_j} \left| \langle h_n, \pi (z)\varphi \rangle \right| ^2 dA(z)\Bigg )^{\frac{1}{2}}\\= & {} \sum _{j=1}^N |\alpha _j|\left| \langle H_{\varOmega -z_j}h_n, h_n\rangle \right| ^{\frac{1}{2}} \le \big (1-e^{-|\varOmega |}\big )^{\frac{1}{2}}. \end{aligned}$$

Finally,

$$\begin{aligned} \int _\varOmega \left| \left( V_{h_0} f\right) (z)\right| ^2 dA(z) = \lim _{k\rightarrow \infty }\int _\varOmega \left| \left( V_{h_0} f_k\right) (z)\right| ^2 dA(z) \le 1-e^{-|\varOmega |}. \end{aligned}$$

$\square$

The next result is a direct consequence of Proposition 2.

Corollary 3

For every $\varepsilon > 0$ there exists $\varOmega$ with infinite Lebesgue measure such that

$$\begin{aligned} \Vert H_\varOmega \Vert \le \varepsilon . \end{aligned}$$