A Wiener Tauberian theorem for operators and functions

We prove variants of Wiener's Tauberian theorem in the framework of quantum harmonic analysis, i.e. for convolutions between an absolutely integrable function and a trace class operator, or of two trace class operators. Our results include Wiener's Tauberian theorem as a special case. Applications of our Tauberian theorems are related to localization operators, Toeplitz operators, isomorphism theorems between Bargmann-Fock spaces and quantization schemes with consequences for Shubin's pseudodifferential operator calculus and Born-Jordan quantization. Based on the links between localization operators and Tauberian theorems we note that the analogue of Pitt's Tauberian theorem in our setting implies compactness results for Toeplitz operators in terms of the Berezin transform. In addition, we extend the results on Toeplitz operators to other reproducing kernel Hilbert spaces induced by the short-time Fourier transform, known as Gabor spaces. Finally, we establish the equivalence of Wiener's Tauberian theorem and the condition in the characterization of compactness of localization operators due to Fern\'andez and Galbis.


Introduction
In operator theory one views the space of trace class operators S 1 as the noncommutative analogue of the space of absolutely integrable functions L 1 (R d ) by viewing the trace of an operator as the substitute of the Lebesgue integral of a function. Over the years this point of view has led to a number of results in operator theory where one has extended concepts for functions to operators in an attempt to formulate operator-theoretic analogues of statements about functions. Guided by this meta-statement, Werner has proposed an operator-theoretic variant of harmonic analysis in [49], which originated from his work in quantum physics and is thus referred to as "quantum harmonic analysis".
In this paper we establish a version of Wiener's Tauberian theorem in the setting of quantum harmonic analysis. Wiener's Tauberian theorem is a cornerstone of harmonic analysis. In short, it analyses the asymptotic properties of a bounded function by testing it with convolution kernels.
Theorem (Wiener's Tauberian Theorem). Suppose f ∈ L ∞ (R d ) and h ∈ L 1 (R d ) with a non-vanishing Fourier transform h. Then the following implication holds for A ∈ C: if then for any g ∈ L 1 (R d ) we have lim x→∞ (g * f )(x) = A defining an operator-theoretic Fourier transform, the Fourier-Wigner transform F W (S) ∈ L ∞ (R 2d ) of a trace class operator S.
The appropriate Fourier transform for functions in L 1 (R 2d ) is the symplectic Fourier transform F σ and the following classes of functions and operators are going to be crucial in our Tauberian theorems for quantum harmonic analysis: W (R 2d ) := {f ∈ L 1 (R 2d ) : F σ (f )(z) = 0 for any z ∈ R 2d }, W := {S ∈ S 1 : F W (S)(z) = 0 for any z ∈ R 2d }.
Our first main result is a generalization of Wiener's Tauberian Theorem for functions on R 2d . Here K denotes the space of compact operators on L 2 (R d ) and I L 2 is the identity operator.
Theorem 4.1 (Tauberian theorem for bounded functions). Let f ∈ L ∞ (R 2d ), and assume that one of the following equivalent statements holds for some A ∈ C: (i) There is some S ∈ W such that f ⋆ S = A · tr(S) · I L 2 + K for some compact operator K ∈ K.
(ii) There is some a ∈ W (R 2d ) such that a(z) dz + h for some h ∈ C 0 (R 2d ).
Then both of the following statements hold: (1) For any T ∈ S 1 , f ⋆ T = A · tr(T ) · I L 2 + K T for some compact operator K T ∈ K. (2) For any g ∈ L 1 (R 2d ), f * g = A · R 2d g(z) dz + h g for some h g ∈ C 0 (R 2d ).
We note that the equivalence (ii) ⇐⇒ (2) is Wiener's original Tauberian theorem. Similarly to Wiener's Tauberian theorem, this theorem concerns the asymptotic properties of the operator R when we use the common intuition that asymptotic properties of an operator are properties that are invariant under compact perturbations, see [6,Chap.3]. There is also a version of the preceding theorem for bounded operators: Theorem 5.1 (Tauberian theorem for bounded operators). Let R ∈ L(L 2 ), and assume that one of the following equivalent statements holds for some A ∈ C: (i) There is some S ∈ W such that R ⋆ S = A · tr(S) + h for some h ∈ C 0 (R 2d ).
(ii) There is some a ∈ W (R 2d ) such that a(z) dz · I L 2 + K for some compact operator K ∈ K. Then both of the following statements hold: (1) For any T ∈ S 1 , R ⋆ T = A · tr(T ) + h T for some h T ∈ C 0 (R 2d ).
(2) For any g ∈ L 1 (R 2d ), R ⋆ g = A · R 2d g(z) dz · I L 2 + K g for some compact operator K g ∈ K.
These Tauberian theorems have numerous applications to localization operators, Toeplitz operators and quantization schemes. The link to localization operators allows us to add another equivalent assumption to Theorem 4.1, formulated in terms of the short-time Fourier transform. Recall that the short-time Fourier transform V φ ψ of ψ for the window φ is given by V φ ψ(z) = ψ, π(z)φ . As condition (ii) in Theorem 4.1 is the condition from Wiener's classical Tauberian theorem, condition (iii) above, which first appeared in the context of localization operators in [24], is a new characterization of the functions to which Wiener's classical Tauberian theorem applies.
It is well-known that T ϕ f and A ϕ,ϕ f are unitarily equivalent. If the window function ϕ is the Gaussian ϕ 0 (x) = e −πx 2 , then V ϕ 0 (L 2 ) is, up to a simple unitary transformation, the space of entire functions on C d known as the Bargmann-Fock space F 2 (C d ), For every F ∈ L ∞ (C d ) one defines the Bargmann-Fock Toeplitz operator T F 2 F on F 2 (C d ) by T F 2 F (H) = P F 2 (F · H) for any H ∈ F 2 (C d ). One has that if f ∈ L ∞ (R 2d ) and F ∈ L ∞ (C d ) are related by F (x + iω) = f (x, −ω) the the following operators are unitarily equvialent: (1) The localization operator A ϕ 0 ,ϕ 0 , the equivalences above allow us to translate statements from convolutions of operators to Toeplitz operators. One example of the interplay of Theorem 4.1, localization operators and Toeplitz operators follows from noting that the theorem gives conditions to ensure that localization operators are compact perturbations of a scaling of the identity, i.e. of the form A · I L 2 + K for 0 = A ∈ C and K ∈ K. Riesz theory yields that if A ϕ,ϕ f = A·I L 2 +K for A = 0 and K ∈ K and A ϕ,ϕ f is injective, then A ϕ,ϕ f is an isomorphism on L 2 (R d ). This implies sufficient conditions for localization operators to be isomorphisms: Proposition 4.10. Let 0 = M ∈ R, a ∈ L ∞ (R 2d ) and ∆ ⊂ R 2d a set of finite Lebesgue measure. Assume that the following assumptions hold: The special case where f = χ Ω such that Ω c has finite measure is of particular interest. Then the localization operator A ϕ,ϕ χ Ω is an isomorphism on L 2 (R 2d ). In particular, any 0 = ψ ∈ L 2 (R d ) is uniquely determined by the values of V ϕ ψ(z) for z ∈ Ω and there exist constants C, D > 0 independent of ψ such that We may translate these results to the polyanalytic Bargmann-Fock space Proposition 4.12.
(1) If Ω ⊂ C d satisfies that Ω c has finite Lebesgue measure, then T is an isomorphism on F 2 n (C d ). Another class of our results concerns the Berezin transform. For the Gabor space V ϕ (L 2 ) we can express the Berezin transform B ϕ : V ϕ (L 2 ) → L ∞ (R 2d ) as a convolution of operators. In particular, the Berezin transform of the Gabor Toeplitz operator T ϕ f is simply a convolution of functions: In particular, this holds for uniformly continuous f . A natural analogue of uniformly continuous functions for operators is the set [11,49]. Werner has obtained the following result in [49] which in light of our Tauberian theorem is an analogue of Pitt's theorem for operators.
Fulsche [28] has recently noted that the preceding theorem implies a result in [10] for the Bargmann-Fock space. We show that the result holds for any Gabor space V ϕ (L 2 ) under certain conditions on ϕ. We would like to stress that it is a Pitt-type theorem for the Tauberian theorem for operators.
Theorem 5.4. Let ϕ ∈ L 2 (R d ) with ϕ L 2 = 1 satisfy that V ϕ ϕ has no zeros, and let T ϕ be the Banach algebra generated by Toeplitz operators T ϕ f ⊂ L(V ϕ (L 2 )) for f ∈ L ∞ (R 2d ). Then the following are equivalent forT ∈ T ϕ .
•T is a compact operator on V ϕ (L 2 ).
Examples of ϕ satisfying that V ϕ ϕ has no zeros were recently investigated in [31], for example the one-sided exponential. Hence these ϕ's give different reproducing kernel Hilbert spaces V ϕ (L 2 ) such that Toeplitz operators are compact if and only if their Berezin transform vanishes at infinity.
The main result in [10] follows in particular, as shown in [28]. We have added a statement on slowly oscillating functions that follows from the original version of Pitt's theorem.
Theorem 5.5 (Bauer, Isralowitz). Let T F 2 be the Banach algebra generated by the Toeplitz operators for a slowly oscillating F ∈ L ∞ (C d ), then the conditions above are equivalent to lim |z|→∞ F (z) = 0.
As a consequence we state a compactness result for Toeplitz operators.
Finally, Theorem 5.2 gives a simple condition for compactness of localization operators in terms of the Gaussian ϕ 0 .
Finally we recall from [45] that any R ∈ L(L 2 ) defines a quantization scheme given by f → f ⋆ R for f ∈ L 1 (R 2d ) and a time-frequency distribution Q R , given by sending ψ ∈ L 2 (R d ) to Q R (ψ)(z) = (ψ ⊗ ψ) ⋆Ř(z) for z ∈ R 2d . The distribution Q R is of Cohen's class since we have Q R (ψ) = aŘ * W (ψ, ψ), where aŘ is the Weyl symbol ofŘ and W (ψ, ψ) the Wigner distribution of ψ.
In the final section we deduce a statement relating compactness properties of the quantization scheme of f → f ⋆ R to properties of Q R (ψ).
Hence if one takes the Gaussian ϕ 0 for (i), then checking if Q R (ϕ 0 ) ∈ C 0 (R 2d ) provides a simple test for checking whether Conditions (iii) and (iv) hold. We apply this result to Shubin's τ -quantization scheme and Born-Jordan quantization.
1.1. Notations and conventions. For topological vector spaces X, Y , we denote by L(X, Y ) the set of continuous, linear operators from X to Y . If X = Y we write L(X ) = L(X, X). The space of compact operators on L 2 (R d ) is denoted by K. For 1 ≤ p < ∞ we let S p denote the Schatten p-class of compact operators with singular values in ℓ p , and we use the convention that S ∞ = L(L 2 ). In particular, S 1 denotes the space of trace class operators on L 2 (R d ), and the trace of a trace class operator T ∈ S 1 is denoted by tr(T ). Also, S 2 is the space of Hilbert-Schmidt operators, which form a Hilbert space with respect to the inner product S, T S 2 = tr(ST * ).
Given a topological vector space X and its continuous dual X ′ , the action of x * ∈ X ′ on y ∈ x is denoted by x * , y X ′ ,X . To agree with the Hilbert space inner product we use the convention that the duality bracket is linear in the first coordinate and antilinear in the second coordinate. The Schwartz functions on R d are denoted by S (R d ).
The Euclidean norm on R d or C d will be denoted by |·|.
For Ω ⊂ R d , χ Ω denotes the characteristic function of Ω. As usual, C 0 (R d ) denotes the continuous functions on R d vanishing at infinity, and we use L 0 (R d ) to denote the space of measurable, bounded functions f on R d such that lim |z|→∞ f (z) = 0,i.e. for every ǫ > 0 there is R > 0 such that |f (z)| < ǫ for a.e. |z| > R. We will refer to L p -spaces on R d , R 2d and C d , and sometimes we will omit explicit reference to the underlying space when it is clear from the context, for instance by writing L(L 2 ) for L(L 2 (R d )). In all statements, measurability and "almost everywhere" properties will refer to Lebesgue measure.

2.1.
Concepts from time-frequency analysis. The mathematical theory of time-frequency analysis will provide the setup and many of the tools we use in this paper. We therefore introduce the time-frequency shifts π(z) ∈ L(L 2 ) for z = (x, ω) ∈ R 2d , given by (π(z)ψ) (t) = e 2πiω·t ψ(x − t) for ψ ∈ L 2 (R d ).
The time-frequency shift π(z) is clearly given as a composition π(z) = M ω T x of a modulation operator M ω ψ(t) = e 2πiω·t ψ(t) and a translation operator The short-time Fourier transform satisfies the important orthogonality relation see [26,29], sometimes called Moyal's identity. Throughout this paper we will use ϕ 0 to denote the normalized Gaussian ϕ 0 (t) = 2 d/4 e −πt 2 for t ∈ R d , and we will often refer to its short-time Fourier transform, which by [29, Lem. 1.5.2] is given by the reader should note already at this point that V ϕ 0 ϕ 0 has no zeros.
2.1.1. Wigner functions and the Weyl transform. Given φ, ψ ∈ L 2 (R d ), a close relative of the short-time Fourier transform V φ ψ is the cross-Wigner distribution W (ψ, φ) defined by The cross-Wigner distribution is the main tool needed to introduce the Weyl transform, which associates to any f ∈ S ′ (R 2d ) an operator L f ∈ L(S (R d ), S ′ (R d )) defined by requiring By the Schwartz kernel theorem [35], . We denote this f by a S , and call it the Weyl symbol of S. In other words, S = L a S . Note that there is no relationship between boundedness of the function f and boundedness of the operator L f on , and there is S ∈ L(L 2 ) such that a S / ∈ L ∞ (R 2d ). See Remark 18 for examples.

Localization operators.
For a mask f ∈ L ∞ (R 2d ) and a pair of windows ϕ 1 , ϕ 2 ∈ L 2 (R d ), we define the localization operator A ϕ 1 ,ϕ 2 f (ψ) ∈ L(L 2 ) by where the integral is interpreted weakly in the sense that we require It is well-known that A ϕ 1 ,ϕ 2 f is bounded on L 2 (R d ) for f ∈ L ∞ (R 2d ) and ϕ 1 , ϕ 2 ∈ L 2 (R d ) [18], but one may also define localization operators for other Banach function spaces of masks f and windows ϕ 1 , ϕ 2 by interpreting the brackets in (4) as duality brackets, see [18]. We postpone this discussion until we have a more suitable framework, which we now introduce.
2.2. Quantum harmonic analysis: convolutions of operators and functions. In this section we introduce the quantum harmonic analysis developed by Werner in [49], the main concepts of which are convolutions of operators and functions and a Fourier transform of operators. For a more detailed introduction in our terminology we refer to [44]. Given any z ∈ R 2d and an operator R ∈ L(L 2 ), we define the translation α z (R) of R by z to be the operator At the level of Weyl symbols, we have that hence α z corresponds to a translation of the Weyl symbol. For f ∈ L 1 (R 2d ) and S ∈ S 1 we then define the convolution f ⋆ S ∈ S 1 by the Bochner integral Hence the convolution of a function with an operator is a new operator. The convolution S ⋆ T of two operators S, T ∈ S 1 is the function S ⋆ T (z) = tr(Sα z (Ť )) for z ∈ R 2d . HereŤ = P T P , with P the parity operator P ψ(t) = ψ(−t). Then S ⋆ T ∈ L 1 (R 2d ) with R 2d S ⋆ T (z) dz = tr(S)tr(T ) and S ⋆ T = T ⋆ S [49]. Taking convolutions with a fixed operator or function is easily seen to be a linear map. One of the most important properties of the convolutions (5) and (6) is that they interact nicely with each other and with the usual convolution f * g(x) = The convolutions also have an interesting interpretation in terms of the Weyl symbol, as we have that As is shown in detail in [44], one can extend the domains of the convolutions by duality. For instance, the convolution f ⋆ S ∈ L( Combining this with a complex interpolation argument gives a version of Young's inequality [44,49]. Recall our convention that S ∞ = L(L 2 ).
, S ∈ S p and T ∈ S q , then f ⋆ T ∈ S r and S ⋆ T ∈ L r (R 2d ) may be defined and satisfy the norm estimates Remark 2. It is worth noting that if S ∈ S 1 and T ∈ L(L 2 ), then S ⋆ T is still given by (6), which can be interpreted pointwise, so that S ⋆ T is a continuous, bounded function.
Young's inequality above shows that the convolutions interact in a predictable way with L p (R 2d ) and S q . We now show that the same is true for functions vanishing at infinity and compact operators. Recall that L 0 (R 2d ) denotes the Banach subspace of L ∞ (R 2d ) consisting of f ∈ L ∞ (R 2d ) that vanish at infinity. The following result shows that convolutions with trace class operators interchange L 0 (R 2d ) and K, which is the basis for our main theorems. These results are known, in particular we mention that part (ii) was proved for rank-one operators S in [12] using essentially the same proof.
Proof. Part (i) is [44,Prop. 4.6]. For (ii) and (iv), note that any f ∈ L 0 (R 2d ) is the limit in in the norm topology of L ∞ (R 2d ) of a sequence of compactly supported Clearly f n ∈ L 1 (R 2d ), hence f n ⋆ S ∈ S 1 ⊂ K. We therefore have by Young's inequality (recall that S ∞ = L(L 2 )): so f ⋆S is the limit in the operator norm of compact operators, hence itself compact. Similarly, f n * a ∈ C 0 (R 2d ) and f n * a converges uniformly to f * a by Young's Finally, (iii) follows by noting that any R ∈ K is the limit in the operator norm of a sequence R n ∈ S 1 of finite-rank operators. Then R n ⋆ a ∈ S 1 is compact, so it follows by (R − R n ) ⋆ a L(L 2 ) ≤ R − R n L(L 2 ) a L 1 that R ⋆ a is the limit in the operator norm of a sequence of compact operators, hence itself compact.
2.2.1. Fourier transforms of functions and operators. As our Fourier transform of functions on R 2d we will use the symplectic Fourier transform F σ , given, for so F σ shares most properties with f : it extends to a unitary operator on L 2 (R 2d ) and to a bijection on S ′ (R 2d ) -see [19]. In addition, F σ is its own inverse: We will also use a Fourier transform of operators, namely the Fourier-Wigner transform F W introduced by Werner [49] (Werner calls it the Fourier-Weyl transform, our usage of Fourier-Wigner agrees with [26]). When S ∈ S 1 , F W (S) is the function As is shown in [45,49], F W extends to a unitary mapping F W : S 2 → L 2 (R 2d ) and a bijection onto The Fourier transforms interact in the expected way with convolutions [49]: if S, T ∈ S 1 and f ∈ L 1 (R 2d ), then We may also connect F W and F σ by the Weyl transform. In fact, we have by [45,Prop. 3.16] that (11) F A main concern for this paper will be functions and operators satisfying that the appropriate Fourier transform never vanishes. Following the notation of [43] for the function case, we introduce the following notation: The key tool for proving the Tauberian theorem for operators is the following generalization of Wiener's approximation theorem, originally proved by Werner [49]. See also [41,44] for more general statements.
(1) The linear span of the translates {α z (S)} z∈R 2d is dense in S 1 .
The special case of rank-one operators. When S ∈ S 1 is a rank-one operator ψ ⊗ φ for ψ, φ ∈ L 2 (R d ), then many of the concepts introduced above are familiar concepts from time-frequency analysis. First we note that by [44, Thm. 5.1], localization operators A ϕ 1 ,ϕ 2 f can be described as convolutions by Other convolutions and Fourier-Wigner transforms of rank-one operators are summarized in the next lemma. See [44, Thm. 5.1 and Lem. 6.1] for proofs. Herě In particular, for ξ, ϕ ∈ L 2 (R d ) Example 2.2 (Standard Gaussian). By (2), F W (ϕ 0 ⊗ϕ 0 )(z) = e −π|z| 2 /2 . We point out this simple case as it shows that ϕ 0 ⊗ ϕ 0 ∈ W. In particular, W is non-empty.

Toeplitz operators and Berezin transforms
In this section we will introduce some families of reproducing kernel Hilbert spaces and the corresponding Toeplitz operators and Berezin transforms. We will relate these spaces and operators to the convolutions introduced in Section 2.2, which will later allow us to deduce results for reproducing kernel Hilbert spaces from the main results this paper. By far the most studied of the spaces we consider is the Bargmann-Fock space F 2 (C d ), and we will later investigate whether some well-known result for F 2 (C d ) can hold for other of the reproducing kernel Hilbert spaces we consider.

Gabor spaces
is an isometry, and one easily confirms that its adjoint operator is where the vector-valued integral is interpreted in a weak sense, see [29,Sec. 3.2] for details. The Gabor space associated with ϕ is then the image V ϕ (L 2 (R d )) ⊂ L 2 (R 2d ), which we denote by V ϕ (L 2 ) for brevity. One can show using (1) that where P Vϕ(L 2 ) denotes the orthogonal projection onto the subspace . By writing out the operators in (13) one deduces that V ϕ (L 2 ) is a reproducing kernel Hilbert space with reproducing kernel (14) k To study such Toeplitz operators in this paper, we will use the map is unitary, Θ ϕ encodes a unitary equivalence, and is easily seen to be a linear, multiplicative and isometric isomorphism. We obtain the following well-known and easily verified result.
Now recall that in a reproducing kernel Hilbert space H consisting of functions on R 2d with normalized reproducing kernel k z for z ∈ R 2d , the Berezin transform BT of a bounded operatorT ∈ L(H) is the function R 2d → C defined by In particular the Berezin transform of the Gabor Toeplitz operator T ϕ f is (14), we have Since Proposition 3.1 and (12) give that we get from the first part that and associativity of convolutions that Remark 4. Gabor spaces and their relation to localization operators has been discussed in [36], with emphasis on f ∈ L ∞ (R 2d ) depending only on x. The reproducing kernel k ϕ z has also been studied as the kernel of determinantal point processes called Weyl-Heisenberg ensembles [4,5].
3.1.1. Gabor spaces with different windows. Having introduced the Gabor spaces V ϕ (L 2 ), we naturally ask whether the properties of V ϕ (L 2 ) as a reproducing kernel Hilbert space depend on the window ϕ in an essential way. As a first result in this direction, we note that the intersection of different Gabor spaces is trivial whenever the windows are not scalar multiples of each other.
as F W is a bijection from S 2 to L 2 (R 2d ). Taking adjoints, we get If we apply (16) to ξ, we obtain Note that dividing by ξ 2 L 2 is allowed, as we assumed V ϕ 1 ξ = 0 which by (1) implies ξ = 0.
Even though the result above shows that Gabor spaces with different windows ϕ 1 and ϕ 2 usually have trivial intersection, there is always an obvious Hilbert space isomorphism Ψ : . However, this does not preserve the reproducing kernels: If we use Proposition 3.1 and Lemma 3.2 to translate parts of Theorem 2.5 into a result on Toeplitz operators, we clearly see that the properties of the window ϕ must be taken into account when studying Toeplitz operators on V ϕ (L 2 ).
The following are equivalent.
). Proof. The result will follow from Theorem 2.5 once we have shown that each statement is equivalent to a statement in that theorem with S = ϕ ⊗ ϕ. As F W (S)(x, ω) = e iπx·ω V ϕ ϕ(x, ω) by Lemma 2.6, (1) states that S ∈ W. Since Proposition 3.1 gives that T ϕ f is unitarily equivalent with A ϕ,ϕ is a bijection, we get that B ϕ is injective if and only if T → T ⋆Š is injective. It is simple to check that the last condition is equivalent to T → T ⋆ S being injective, as a calculation shows that T ⋆Š(z) =Ť ⋆ S(−z).
Remark 5. The other parts of Theorem 2.5 could also be translated into statements on V ϕ (L 2 ), and one could obtain other equivalences by imposing weaker requirements on the set of zeros of V ϕ ϕ, see [41,44].

3.2.
Toeplitz operators on Bargmann-Fock space. For the Gaussian ϕ 0 , the Gabor space V ϕ 0 (L 2 ) is closely related to another much-studied reproducing kernel Hilbert space: the Bargmann-Fock space F 2 (C d ), consisting of all analytic func- An important tool in the study of F 2 (C d ) is the Bargmann transform, which is the unitary mapping B : where A : L 2 (R 2d ) → L 2 (C d , e −π|z| 2 dz) is a unitary operator given by , as it may be written as the composition B • V * ϕ 0 | Vϕ 0 (L 2 ) of unitary operators. Hence A allows us to relate the spaces V ϕ 0 (L 2 ) and F 2 (C d ).
The orthogonal projection from L 2 (C d , e −π|z| 2 dz) to F 2 (C d ) is given by For our purposes it is convenient to note that we can use the reproducing kernel k ϕ 0 (x,ω) for V ϕ 0 (L 2 ) to express K z for z = x + iω by as follows from the calculation Using (18) and the unitarity of A, one can calculate that if f ∈ L ∞ (R 2d ) and F ∈ L ∞ (C d ) are related by (20) F In combination with Proposition 3.1 this gives the following result. (20). Then the following operators are unitarily equivalent.
(1) The localization operator . Remark 6. The simple result above is far from new, going back to at least [17]. A related and more complicated question that appears in the literature is to relate A ϕ,ϕ f , where ϕ needs no longer be Gaussian, to a Bargmann-Fock Toeplitz operator T F 2 (I+D)F , where D is some differential operator [2,17,22]. The Berezin transform can also be defined on F 2 (C d ). Since A : V ϕ (L 2 ) → F 2 (C) is unitary, one easily checks using (19) that the normalized reproducing kernelk z on F 2 (C d ) is This implies the following result on the Berezin transform B F 2 on F 2 (C d ).

That this last expression equals (B
For the formula for Toeplitz operators, combine the first part with (21) and the final part of Lemma 3.2.
The results above show the intimate connection between F 2 (C d ) and the Gabor space V ϕ 0 (L 2 ). Many of the results known for F 2 (C d ) can easily be translated into results for V ϕ 0 (L 2 ), and we will later investigate certain conditions on ϕ that allow us to generalize these results to other Gabor spaces V ϕ (L 2 ). (17), we may identify V ϕ 0 (L 2 ) and the Bargmann-Fock space by the operator A : L 2 (R 2d ) → L 2 (C d , e −π|z| 2 dz). If the Gaussian ϕ 0 is replaced by another Hermite function ϕ n for n ∈ N d , and we define the polyanalytic Bargmann transform B n :

Polyanalytic Bargmann-Fock spaces. By
then the image of B n , which we denote by F 2 n , is again a reproducing kernel Hilbert space with reproducing kernel K ϕn z for z = x + iω given by K ϕn z (x ′ + iω ′ ) = e iπx·ω e π|z| 2 /2 AK ϕn (x,−ω) (x ′ + iω ′ ). Unlike the Bargmann-Fock space F 2 = F 2 0 , F 2 n does not in general consist of analytic functions, but rather of so-called polyanalytic functions. For this reason F 2 n is sometimes called the true polyanalytic Fock space of degree n [1, 3,8]. Following [40] we define, given F ∈ L ∞ (C d ), the polyanalytic Toeplitz operator T n . Similarly to Bargmann-Fock space the orthogonal projection P F 2 n from L 2 (C d , e −π|z| 2 dz) to F 2 n is given by If f ∈ L ∞ (R 2d ) and F ∈ L ∞ (C d ) are related as in (20), one can show that T ϕn Hence we obtain the following result. Proposition 3.7. Let f ∈ L ∞ (R 2d ) and F ∈ C d be related as in (20). For n ∈ N d , the following operators are unitarily equivalent.
(1) The localization operator A ϕn,ϕn (3) The polyanalytic Toeplitz operator T We have related polyanalytic Toeplitz operators to Gabor Toeplitz operators on V ϕn (L 2 ). By [38, (4.16)], V ϕn ϕ n has zeros if and only if n = 0. An easy argument using the previous proposition then translates Proposition 3.4 into the following statement.
if and only if n = 0. In other words, assigning a bounded function to a Toeplitz operator is only injective on the Bargmann-Fock space.

A Tauberian theorem for bounded functions
As our first main result we present a generalization of Wiener's classical Tauberian theorem that applies to bounded functions and convolutions with integrable functions and trace class operators. The key tool is Werner's generalization of Wiener's approximation theorem from Theorem 2.5.
Theorem 4.1 (Tauberian theorem for bounded functions). Let f ∈ L ∞ (R 2d ), and assume that one of the following equivalent statements holds for some A ∈ C: (i) There is some S ∈ W such that f ⋆ S = A · tr(S) · I L 2 + K for some compact operator K ∈ K. (ii) There is some a ∈ W (R 2d ) such that Then both of the following statements hold: (1) For any T ∈ S 1 , f ⋆ T = A · tr(T ) · I L 2 + K T for some compact operator K T ∈ K. (2) For any g ∈ L 1 (R 2d ), f * g = A · R 2d g(z) dz + h g for some h g ∈ C 0 (R 2d ).
Proof. We start by proving that (i) and (ii) are equivalent. Assume (i), and consider a = S ⋆ S ∈ L 1 (R 2d ). Since F σ (S ⋆ S)(z) = F W (S)(z) 2 for any z ∈ R 2d by (9), we obtain both that F σ (a) has no zeros and (by evaluating the relation at z = 0) that R 2d a(z) dz = tr(S) · tr(S).
Then observe using associativity of the convolutions that a(z) dz + K ⋆ S, and K ⋆ S ∈ C 0 (R 2d ) by Lemma 2.3. The proof that (ii) implies (i) is similar by picking S = a ⋆ T , where T ∈ S 1 is any operator in W. Then F W (S)(z) = F σ (a)(z)F W (T )(z) by (10), so F W (S) has no zeros and tr(S) = R 2d a(z) dz · tr(T ) by evaluating the relation at z = 0. Furthermore, associativity of convolutions gives a(z) dz · tr(T ) · I L 2 + h ⋆ T by Lemma 2.4 = A · tr(S) · I L 2 + h ⋆ T, and h ⋆ T ∈ K by Lemma 2.3. Hence (i) and (ii) are equivalent. The fact that (ii) implies (2) is Wiener's classical Tauberian theorem. The proof will therefore be completed if we can show (i) =⇒ (1), so assume that S satisfies (i), and for now assume A = 0. In short, we assume f ⋆ S ∈ K. We need to show that f ⋆ T ∈ K for any T ∈ S 1 . Part (3) of Theorem 2.5 implies that T is the limit in the norm of S 1 of a sequence r n ⋆ S for r n ∈ L 1 (R 2d ). By commutativity and associativity of the convolutions, f ⋆ (r n ⋆ S) = r n ⋆ (f ⋆ S) ∈ K by Lemma 2.3.

Proposition 2.2 then gives that
Hence f ⋆ T is the limit in the operator norm of compact operators, thus compact. Finally, assume that A = 0. Then (f − A) ⋆ S ∈ K by Lemma 2.4, so the result for A = 0 implies that (f − A) ⋆ T ∈ K for any T ∈ S 1 , and applying Lemma 2.4 again we see that this is equivalent to (1).
The case A = 0 is particularly interesting, as it concerns the compactness of operators of the form f ⋆ T for T ∈ S 1 . We will return to this special case on several occasions.
(1) Note that the convolution of a bounded and an integrable function is continuous, so we lose no generality by assuming that h and h g belong to C 0 (R 2d ) rather than merely assuming that they belong to L 0 (R 2d ).
(2) As already mentioned in the proof, the classical Tauberian theorem of Wiener is the implication (ii) =⇒ (2). (3) The conditions on the Fourier transforms of S in (i) are necessary to imply (1) and (2). To see this, assume that S ∈ S 1 satisfies F W (S)(z 0 ) = 0 for some z 0 = (x 0 , ω 0 ) ∈ R 2d . Then consider the function f z 0 (z) = e 2πiσ(z 0 ,z) ∈ L ∞ (R 2d ). One can show that for any T ∈ S 1 we have In particular, f z 0 ⋆ S = 0 ∈ K since F W (S)(z 0 ) = 0, so apart from the condition on F W (S) we see that S satisfies (i) with A = 0. However, is not true for f z 0 . A similar argument with the same functions f z 0 shows that the condition on a in (ii) is also necessary.

A result by Fernández and Galbis.
In [24], Fernández and Galbis proved the following result on compactness of localization operators. (1) This requirement is weaker than both f ∈ L 0 (R 2d ) and V Φ f ∈ C 0 (R 4d ) for some non-zero Φ ∈ S (R 2d ). Proving that either of these two statements implies compactness of A ϕ 1 ,ϕ 2 f requires far less advanced tools than (22), see [24].
and certain distributions such as Dirac's delta distribution, see [29].
This allows us to add another equivalent assumption to Theorem 4.1, formulated in terms of the short-time Fourier transform of f . Proof. Consider the operator S = ϕ 0 ⊗ ϕ 0 . Then S ∈ W by (2) and f ⋆ S = A ϕ 0 ,ϕ 0 f by (12). If (iii) is satisfied, Theorem 4.2 implies using Lemma 2.4 that is compact for any ϕ 1 , ϕ 2 ∈ S (R d ), so Theorem 4.2 implies that (iii) holds.
Remark 9. One may easily calculate that Condition (iii) therefore says that for any R > 0, if fixed x is picked with |x| sufficiently large, then  • There is some non-zero Φ ∈ S (R 2d ) such that for every R > 0 • There is a ∈ W (R 2d ) and h ∈ C 0 (R 2d ) such that Remark 10. One might naturally ask if this result holds for R d for any d ≥ 1, and not just for even d. Our proof exploits Theorem 4.2, which has no analogue for odd d. We can therefore not extend the proof to the general case.
Then F σ (a) = a has no zeros, and Rather surprisingly, we may prove (1) directly in this case by considering the operator side of our setup. For any T ∈ S 1 , we obtain that F W (T ) ∈ L 2 (R 2d ) since S 1 ⊂ S 2 and F W : The key to this calculation is the inclusion L ∞ (R 2d )· F W (S 1 ) ⊂ L 2 (R 2d ) -the corresponding function result L ∞ · F σ (L 1 ) ⊂ L 2 is not true by the results in [16].
The examples above show that it is not necessary to have lim |z|→∞ f (z) = 0 in order to satisfy assumptions (i) and (ii) with A = 0. A well-known result in the Tauberian theory of functions due to Pitt [46] says that if we assume that f is slowly oscillating, then lim |z|→∞ f (z) = 0 is necessary for f to satisfy (ii).
Remark 11. Any uniformly continuous f ∈ L ∞ (R 2d ) is slowly oscillating, hence if such f satisfies (i) with A = 0, then f ∈ C 0 (R 2d ). This weaker statement actually follows from the correspondence theory introduced by Werner in [49], more precisely by [49,Thm. 4.1 (3)]. In Werner's terminology C 0 (R 2d ) and K are corresponding subspaces, since convolutions with trace class operators interchanges these two spaces by Lemma 2.3. We will see the operator-analogue of this result in Section 5.1 4.2.1. Consequences for Toeplitz operators. We now formulate a version of the Tauberian theorem for (polyanalytic) Bargmann-Fock Toeplitz operators. As a preliminary observation, let H 1 , H 2 be two Hilbert spaces. If S ∈ L(H 1 ) and T ∈ L(H 2 ) are unitarily equivalent, i.e. there is unitary U : H 1 → H 2 such that S = U * T U, then one easily checks that S = A · I H 1 + K 1 for A ∈ C and compact K 1 ∈ L(H 1 ) if and only if T = A · I H 2 + K 2 for compact K 2 ∈ L(H 2 ).
The following are equivalent: Furthermore, if any of the equivalent conditions above holds, then for any n ∈ N d the polyanalytic Toeplitz operator T n +K n , whereK n is a compact operator on F 2 n (C d ). Proof. By Proposition 3.5, T F 2 F is unitarily equivalent to A ϕ 0 ,ϕ 0 f = f ⋆ (ϕ 0 ⊗ ϕ 0 ). By the remark above, part (i) holds if and only if f ⋆ (ϕ 0 ⊗ ϕ 0 ) = A · I L 2 + K 0 for some compact operator K 0 on L 2 (R d ). Since ϕ 0 ⊗ ϕ 0 ∈ W by (2), the fact that (i), (ii) and (iii) are equivalent follows from Proposition 4.3.
As we have seen that (i) implies that f ⋆ (ϕ 0 ⊗ ϕ 0 ) = A · I L 2 + K 0 and that ϕ 0 ⊗ ϕ 0 ∈ W, Theorem 4.1 implies that for every h there is a compact K n with f ⋆ (ϕ n ⊗ ϕ n ) = A · I L 2 · tr(ϕ n ⊗ ϕ n ) + K n = A · I L 2 + K n .
The last statement then follows as T The same reasoning gives the following Tauberian theorem for Toeplitz operators on Gabor spaces using Proposition 3.1.
Proposition 4.7. Let f ∈ L ∞ (R 2d ) and A ∈ C. The following are equivalent.
(i) There is some ϕ ∈ L 2 (R d ) such that V ϕ ϕ has no zeros and T ϕ f = A·I Vϕ(L 2 ) +K for some compact operator K ∈ L(V ϕ (L 2 )) .
(ii) There is some a ∈ W (R 2d ) such that f * a = A · R 2d a(z) dz + h for some h ∈ C 0 (R 2d ). (iii) There is some non-zero Φ ∈ S (R 2d ) such that for every R > 0 Furthermore, if any of the equivalent conditions above holds, then for every nor-

4.3.
Injectivity of localization operators and Riesz theory of compact operators. Theorem 4.1 gives conditions to ensure that a localization operator A ϕ,ϕ f is a compact perturbation of a scaling of the identity, i.e. of the form A·I L 2 +K for 0 = A ∈ C and K ∈ K. The theory of such operators, sometimes referred to as Riesz theory due to the seminal work of F. Riesz [47], contains several powerful results similar to those that hold for matrices. We will use the following result, see [15, Lem. 6.30 & Thm. 6.33] for proofs. Proposition 4.8. Assume that T ∈ L(L 2 ) is of the form A · I L 2 + K for A = 0 and K ∈ K. Then T has closed range and dim(ker T ) = dim(coker(T )) < ∞. In particular, T is injective if and only if T is surjective.
As an obvious consequence, we note that if A ϕ,ϕ f = A · I L 2 + K for A = 0 and K ∈ K and A ϕ,ϕ f is injective, then A ϕ,ϕ f is an isomorphism on L 2 (R d ). Inspired by this, we investigate conditions ensuring that localization operators are injective. The proof of the next result is similar to that of [14,Lem. 1.4]. Lemma 4.9. Assume that f ∈ L ∞ (R 2d ) such that f (z) ≥ 0 for a.e. z ∈ R 2d .
We deduce sufficient conditions for localization operators to be isomorphisms. Proposition 4.10. Let 0 = M ∈ R, a ∈ L ∞ (R 2d ) and ∆ ⊂ R 2d a set of finite Lebesgue measure. Assume that the following assumptions hold: Proof. By Lemma 4.9 part (1), A ϕ,ϕ f is injective. By assumption (iii), Theorem 4.1 gives that a ⋆ (ϕ ⊗ ϕ) ∈ K, so that is a compact perturbation of a scaling of the identity. Hence Proposition 4.8 implies that A ϕ,ϕ f is also surjective.
(1) Finding specific examples of a satisfying the assumptions above is not difficult, but it is worth noting that a need not vanish at infinity. For instance, a standard construction gives continuous a ∈ L 1 (R 2d ) ∩ L ∞ (R 2d ) such that 0 ≤ a ≤ 1, lim sup |z|→∞ |a(z)| = 1 and lim inf |z|→∞ |a(z)| = 0. Then a satisfies all three conditions above for M > 0, even though f = M + a has no limit as |z| → ∞. Of course, if we add the condition that a is slowly oscillating, then a must vanish at infinity by Theorem 4.5.
We state a special case of Proposition 4.10 as a theorem, namely the case where f = χ Ω such that Ω c has finite measure. We find that as long as Ω c has finite measure, the values of V ϕ ψ(z) for z ∈ Ω c are not needed to reconstruct ψindependently of the geometry of Ω and the window ϕ.
Theorem 4.11. Assume that Ω ⊂ R 2d satisfies that Ω c has finite Lebesgue measure, and that 0 = ϕ ∈ L 2 (R d ). Then the localization operator A ϕ,ϕ χ Ω is an isomorphism on L 2 (R 2d ). In particular, any 0 = ψ ∈ L 2 (R d ) is uniquely determined by the values of V ϕ ψ(z) for z ∈ Ω and there exist constants C, D > 0 independent of ψ such that Proof. This is a special case of Proposition 4.10 with M = 1 and a = −χ Ω c . Then f = 1 − χ Ω c = χ Ω , and one easily checks that the conditions in the proposition are satisfied with ∆ = Ω c , in particular (iii) follows as χ Ω c ∈ L 1 (R 2d ).
Remark 13. If ϕ belongs to Feichtinger's algebra M 1 (R d ) [23,29], then invertibility of A ϕ,ϕ f on L 2 (R d ) implies that A ϕ,ϕ f is also invertible on all modulation spaces M p,q (R d ) for 1 ≤ p, q ≤ ∞ (see [29] for an introduction to modulation spaces). This follows by combining [ Hence the results of this section may be translated into results for Toeplitz operators on F 2 n (C d ). We include a couple of such results in the next statement. One may of course obtain isomorphism results for Gabor spaces in the same way by using Proposition 3.1.

Proposition 4.12.
(1) If Ω ⊂ C d satisfies that Ω c has finite Lebesgue measure, then T F 2 n χ Ω is an isomorphism on F 2 n (C d ).
(2) There is a real-valued, continuous F ∈ L ∞ (C d ) such that lim |z|→∞ |F (z)| does not exist, yet T is an isomorphism on F 2 n (C d ). Proof. In light of Proposition 3.7, the first part follows from Theorem 4.11 and the second from Remark 12.

A Tauberian theorem for bounded operators
A guiding principle in the theory of quantum harmonic analysis is that the role of functions and operators may often be interchanged in theorems. It should therefore come as no surprise that we can prove a Tauberian theorem where the bounded function f from Theorem 4.1 is replaced by a bounded operator R, with just a few modifications of the proof.
Theorem 5.1 (Tauberian theorem for bounded operators). Let R ∈ L(L 2 ), and assume that one of the following equivalent statements holds for some A ∈ C: (i) There is some S ∈ W such that R ⋆ S = A · tr(S) + h for some h ∈ C 0 (R 2d ).
(ii) There is some a ∈ W (R 2d ) such that a(z) dz · I L 2 + K for some compact operator K ∈ K. Then both of the following statements hold: (1) For any T ∈ S 1 , R ⋆ T = A · tr(T ) + h T for some h T ∈ C 0 (R 2d ).
(2) For any g ∈ L 1 (R 2d ), R ⋆ g = A · R 2d g(z) dz · I L 2 + K g for some compact operator K g ∈ K.
Proof. The equivalence of the assumptions is proved in a similar way as for Theorem 4.1: for (i) =⇒ (ii) pick a = S ⋆ S, and for (ii) =⇒ (i) pick S = a ⋆ T for any T ∈ W.
Then assume that (i) holds with A = 0, the extension to A = 0 is done as in the proof of Theorem 4.1. To show (1), one proceeds as in the proof of Theorem 4.1 by first showing that S ⋆ T ∈ C 0 (R 2d ) if T = r ⋆ S for some r ∈ L 1 (R 2d ). Using Theorem 2.5 one has that any T ∈ S 1 is the limit in the norm of S 1 of a sequence r n ⋆ S with r n ∈ L 1 (R 2d ). The proof is completed by showing that the sequence R ⋆ (r n ⋆ S) -which is a sequence of functions in C 0 (R 2d ) -converges uniformly to R ⋆ T . Since C 0 (R 2d ) is closed under uniform limits, this implies (1).
The proof that (i) implies (2) follows the same pattern. First show it for g = T ⋆S for some T ∈ S 1 , then extend to all g by density, since Theorem 2.5 implies that any g ∈ L 1 (R 2d ) is the limit of a sequence T n ⋆ S for T n ∈ S 1 .
Hence the condition in (i) is necessary. To show that the condition on a in (ii) is necessary one uses a similar argument and the fact that Example 5.1. If R ∈ L(L 2 ) satisfies that F W (R) ∈ L ∞ (R 2d ), then R satisfies assumption (ii) of Theorem 5.1 with A = 0 -such R are the operator-analogues of the pseudomeasures considered in Example 4.1. To prove this, let S = ϕ 0 ⊗ ϕ 0 . Then F W (S)(z) = e −π|z| 2 , so S ∈ W, and By Fourier inversion we have which belongs to C 0 (R 2d ) by the Riemann-Lebesgue lemma.
An example of such R is R = P , the parity operator. One can show that F W (P )(z) = 2 d for any z ∈ R 2d , hence P is a non-compact operator satisfying assumption (ii) of Theorem 5.1 with A = 0. We will return to this and other examples below. 5.1. Pitt improvements and characterizing compactness using Berezin transforms. As we saw in Theorem 4.5, Pitt's classical theorem gives a condition on f ∈ L ∞ (R 2d ) that ensures that In particular, we noted that this is true if f is uniformly continuous. To generalize this statement to operators R ∈ L(L 2 ), recall that f ∈ L ∞ (R 2d ) is uniformly continuous if and only if z → T z (f ) is continuous map from R 2d to L ∞ (R 2d ). Hence a natural analogue of the uniformly continuous functions is the set this heuristic was also followed by Werner [49] and Bekka [11]. With this in mind, the following result from [49] is an analogue of Pitt's theorem for operators.
Theorem 5.2. Let R ∈ C 1 . The following are equivalent.
Proof. That the first statement implies the other two is Lemma 2.3. That the other statements imply the first follows from the theory of corresponding subspaces developed by Werner in [49], more precisely from [49,Thm. 4.1 (3)]. In the notation of [49] we have picked D 0 = C 0 (R 2d ) and D 1 = K.
We then try to gain a better understanding of the elements of C 1 .
Furthermore, it is not difficult to see that C 1 equipped with the operator norm is a Banach algebra. Hence it must contain the Banach algebra generated by elements of the form f ⋆ T for f ∈ L ∞ (R 2d ) and T ∈ S 1 , and Proposition 5.2 applies to operators in this Banach algebra.
This allows us to apply the results above to characterizing compactness of Toeplitz operators by their Berezin transform, a much-studied question going back to results of Axler and Zheng [7] for the so-called Bergman space, and soon after Engliš [21] for the Bargmann-Fock space F 2 (C d ). The central question is whether a Toeplitz operator on a reproducing kernel Hilbert space must be compact if its Berezin transform vanishes at infinity -see Section 4 of [9] for an overview over results of this nature in the literature. We will use Proposition 5.2 to reprove the main result of [10] for F 2 (C d ) and extend it to a class of Gabor spaces, but we hasten to add that the method of proving the results of [10] using the results of [49] was already noted recently by Fulsche [28]. Before the proof, recall the linear and multiplicative isometric isomorphism Θ ϕ : L(V ϕ (L 2 )) → L(L 2 ) from (15), which satisfies that Θ ϕ (T ϕ f ) = A ϕ,ϕ f and B ϕT = Θ ϕ (T ) ⋆ (φ ⊗φ).
Theorem 5.4. Let ϕ ∈ L 2 (R d ) with ϕ L 2 = 1 satisfy that V ϕ ϕ has no zeros, and let T ϕ be the Banach algebra generated by Toeplitz operators T ϕ f ⊂ L(V ϕ (L 2 )) for f ∈ L ∞ (R 2d ). Then the following are equivalent forT ∈ T ϕ .
Proof. First note that the assumption on V ϕ ϕ means that ϕ ⊗ ϕ ∈ W by Lemma 2.6, and as a simple calculation shows that F W (φ ⊗φ)(z) = F W (ϕ ⊗ ϕ)(−z) it also means thatφ ⊗φ ∈ W. To see that the first statement implies the second, note that Θ ϕ (T ) is compact if and only ifT is, so by Lemma 2.3. For the other direction, it is clear by the properties of Θ ϕ that it maps T ϕ into the Banach algebra generated by localization operators A ϕ,ϕ f = f ⋆ (ϕ ⊗ ϕ) for f ∈ L ∞ (R 2d ). In particular, Θ ϕ (T ϕ ) ⊂ C 1 by (23) as C 1 is a Banach algebra containing A ϕ,ϕ f for all f ∈ L ∞ (R 2d ). Since B ϕT = Θ ϕ (T ) ⋆ (φ ⊗φ) ∈ C 0 (R 2d ) andφ ⊗φ ∈ W by assumption, Proposition 5.2 gives that Θ ϕ (T ) is compact, henceT is compact as Θ ϕ is a unitary equivalence by definition.
The last statement follows from Theorem 4.5, as T ϕ f is compact if and only if Remark 15. Similar techniques have also recently been used by Hagger [34] to give a characterization of some generalizations of T ϕ .
There are several examples of ϕ satisfying that V ϕ ϕ has no zeros, which by the proposition gives examples of reproducing kernel Hilbert spaces V ϕ (L 2 ) such that Toeplitz operators are compact if and only if their Berezin transform vanishes at infinity. One example is the one-sided exponential ϕ(t) = χ [0,∞) (t)e −t for t ∈ R considered by Janssen [37], and new examples were recently explored in [31].
Essentially the same argument as for Theorem 5.4, only replacing Θ ϕ by the map Θ F 2 : L(F 2 (C d )) → L(L 2 ) defined by Θ F 2 (T ) = B * T B, gives a Bargmann-Fock space result from [10]. For this to work, it is important that ϕ 0 ⊗ ϕ 0 ∈ W, since Proposition 3.5 and Lemma 3.6 relate the Bargmann-Fock setting to convolutions with ϕ 0 ⊗ ϕ 0 . The last remark on slowly oscillating functions is, to our knowledge, a new contribution, and follows from Theorem 4.5. The definition of slowly oscillating functions on R 2d given after that theorem is adapted to C d in an obvious way.
Theorem 5.5 (Bauer, Isralowitz). Let T F 2 be the Banach algebra generated by the Toeplitz operators T F 2 F for F ∈ L ∞ (C d ). The following are equivalent forT ∈ T F 2 . •T is a compact operator on F 2 (C d ).
for a slowly oscillating F ∈ L ∞ (C d ), then the conditions above are equivalent to lim |z|→∞ F (z) = 0.
By Lemma 3.6 we immediately obtain the following compactness criterion.
Remark 16. One could also define the Berezin transform for Toeplitz operators on polyanalytic Bargmann-Fock spaces and relate it to convolutions with ϕ n ⊗ ϕ n . However, we would not be able to apply Proposition 5.2 to this case, as V ϕn ϕ n always has zeros for n = 0.

Quantization schemes and Cohen's class
The perspective of [45] is that any R ∈ L(L 2 ) defines both a quantization scheme and a time-frequency distribution. The quantization scheme associated with Rby which we simply mean a map sending functions on phase space R 2d to operators on L 2 (R d ) -is given by The time-frequency distribution Q R associated with R is given by sending ψ ∈ L 2 (R d ) to its time-frequency distribution Recall that a quadratic time-frequency distribution Q is said to be of Cohen's class if there is some a ∈ S ′ (R 2d ) such that (24) Q(ψ) = a * W (ψ, ψ) for all ψ ∈ S (R d ).
The distribution Q R is of Cohen's class as (7) implies that where aŘ is the Weyl symbol ofŘ. Using Theorem 5.1, we deduce the following result relating compactness of the quantization scheme of R to C 0 (R 2d ) membership of Q R .
Proposition 6.1. Let R ∈ L(L 2 ). The following are equivalent.
The gist of the above proposition is that (i) provides a simple test for checking whether (iii) and (iv) hold. A typical choice for ϕ in (i) would be the Gaussian ϕ = ϕ 0 , then Q R (ϕ 0 ) is the so-called Husimi function of R. Hence the quantization f ⋆ R of any f ∈ L 1 (R 2d ) is compact and Q R (ψ) ∈ C 0 (R 2d ) for any ψ ∈ L 2 (R d ) if and only if the Husimi function of R belongs to C 0 (R 2d ). if τ = 1 2 , where δ 0 is Dirac's delta distribution. A slightly tedious calculation using the definition (3) shows that the Weyl transform S τ of a τ is given for ψ ∈ S (R d ) by as already noted for d = 1 in [42,Thm. 7.2]. If τ ∈ (0, 1), it is easy to check that S τ is bounded on L 2 (R d ) with S τ L(L 2 ) = 1 (1−τ ) d/2 τ d/2 , that S * τ = S 1−τ ,Š τ = S τ and the inverse of S τ is τ d (1 − τ ) d S 1−τ . In particular, S τ is not compact.
In light of (25), [13,Prop. 5.6] states that Q Sτ (ψ) is the τ -Wigner distribution W τ (ψ) introduced in [13], given explicitly by On the other hand, we easily find for f ∈ L 1 (R 2d ) and ψ ∈ S (R d ) that In the last line we use that Q S (ψ) = Q S * (ψ) for S ∈ L(L 2 ), and S * τ = S 1−τ . This shows precisely that f ⋆ S 1−τ satisfies the definition of the τ -Weyl quantization of f introduced by Shubin [48] -in the notation of [13] we have that f ⋆ S 1−τ = W f τ . The case τ = 1/2 is of particular interest, as S 1/2 = S 1−1/2 = 2 d P -a scalar multiple of the parity operator. This case corresponds to the Weyl calculus, in the sense that Q 2 d P (ψ) = W (ψ, ψ) for ψ ∈ L 2 (R d ) and f ⋆ (2 d P ) is the Weyl transform of f for f ∈ L 1 (R 2d ).
We we can now show that the τ -Wigner theory and the non-compact operators S τ give a family of non-trivial examples to Theorem 5.1. The compactness part of the next result was also noted using different methods in [13, Thm. 6.9]. Proposition 6.2. Let τ ∈ (0, 1). Then S τ satisfies the assumptions of Theorem 5.1 with A = 0, and (1) W τ (ψ) = Q Sτ (ψ) ∈ C 0 (R 2d ) for any ψ ∈ L 2 (R d ).
Remark 18. The operators S 0 and S 1 are clearly not bounded on L 2 (R d ), even though a 0 , a 1 ∈ L ∞ (R 2d ). Hence a 0 and a 1 are examples of bounded functions with unbounded Weyl transform. Similarly, S 1/2 is a bounded operator with unbounded Weyl symbol.
We end by considering the example of Born-Jordan quantization.
The associated quantization scheme f → S BJ is then the Born-Jordan quantization [20]. For d = 1 it was shown in [42,Prop. 2] that S BJ ∈ L(L 2 ). Since (26) shows that F W (S BJ ) ∈ L ∞ (R 2d ), combining Example 5.1 and Proposition 6.1 we may conclude that the Born-Jordan quantization of any f ∈ L 1 (R 2 ) is compact, and that the Born-Jordan distribution of any ψ ∈ L 2 (R) belongs to C 0 (R 2 ). 6.1. Counterexample to a Schatten class version of Theorem 5.1. For the special case A = 0, Theorem 5.1 states that if R ⋆ a ∈ K for some a ∈ W (R 2d ), then R ⋆ g ∈ K for all g ∈ L 1 (R 2d ). An obvious generalization is to replace K by a Schatten class S p for some 1 ≤ p < ∞. Is it true that R ⋆ a ∈ S p for a ∈ W (R 2d ) implies that R ⋆ g ∈ S p for all g ∈ L 1 (R 2d )? A simple counterexample is provided by the Weyl calculus. Example 6.2. Recall that S 1/2 ⋆ f is the Weyl transform of f ∈ L 1 (R 2d ). If we let a(z) = 2 d e −π|z| 2 , then a ∈ W (R 2d ) and it is well-known that the Weyl transform a ⋆ S 1/2 of a is the rank-one operator ϕ 0 ⊗ ϕ 0 . In particular, a ⋆ S 1/2 ∈ S 1 ⊂ S p for any 1 ≤ p ≤ ∞. However, if we pick f ∈ L 1 (R 2d ) \ L 2 (R 2d ), then f ⋆ S 1/2 / ∈ S p for any 1 ≤ p ≤ 2, since the Weyl transform is a unitary mapping from L 2 (R 2d ) to S 2 , and S p ⊂ S 2 for 1 ≤ p ≤ 2. Hence we cannot conclude from a ⋆ S 1/2 ∈ S p for a ∈ W (R 2d ) that f ⋆ S 1/2 ∈ S p for all f ∈ L 1 (R 2d ), at least for 1 ≤ p ≤ 2.