Polyanalytic Toeplitz operators: isomorphisms, symbolic calculus and approximation of Weyl operators

We discuss an extension of Toeplitz quantization based on polyanalytic functions. We derive isomorphism theorem for polyanalytic Toeplitz operators between weighted Sobolev-Fock spaces of polyanalytic functions, which are images of modulation spaces under polyanalytic Bargmann transforms. This generalizes well-known results from the analytic setting. Finally, we derive an asymptotic symbol calculus and present an asymptotic expansion of complex Weyl operators in terms of polyanalytic Toeplitz operators.


Introduction
Bargmann transforms and Fock spaces provide a widely used language that connects the theory of entire functions with a variety of topics in theoretical and applied mathematics, including signal analysis, quantum mechanics as well as complex geometry and analytic microlocal analysis.
This area of mathematics goes back to the seminal work [Bar61] of Bargmann that has been motivated by applications in quantum mechanics. In microlocal analysis, generalized Bargmann transforms are mostly known as Fourier-Bros-Iagolitzer transforms and were first applied by Bros, Iagolnitzer and Stapp in order to analyze wave front sets, see e.g. [IS69], or [Sjö82] for a more recent and general approach. Janssen established the link between the Bargmann transform and Gabor frames in [Jan82] which allowed him to apply methods from complex analysis to problems in signal analysis. This connection between Gabor frames and complex analysis has turned out to be very fruitful, e.g. for the characterization of the Gabor frame set of a Gaussian in [Lyu92,Sei92], or the construction of unconditional bases for Bargmann-Fock spaces in [FGW92].
Toeplitz operators provide a natural framework to describe linear transformations in Fock type spaces that can be interpreted as signal manipulations, quantum observables or pseudodifferential operators. In fact, Toeplitz operators are nothing else but the image of anti-Wick or localization operators under the Bargmann transform. Putting it differently, localization operators are in fact Toeplitz operators on the phase space, see [BCG04,Eng09,AF15].
The aim of this paper is to lift the well-established theory of Toeplitz operators to the polyanalytic setting, following initial works of Abreu, Gröchenig and Faustino [Abr10,AG10,Fau11] as well as [Gof10,EZ17,RV19]. That is, we introduce multiplication operators on Bargmann-Fock type spaces of polyanalytic functions and, thus, provide a whole new family of quantization schemes. Polyanalytic Toeplitz operators appear as the natural complexification of localization operators with Hermite function windows. Moreover, polyanalytic Bargmann-Fock spaces are precisely the images of the classical modulation spaces under polyanalytic Bargmann transforms. Polyanalytic functions were first studied by Kolossov more than a century ago. Howoever, it was not until the seminal work of Vasilevski [Vas00] that this generalization of analytic functions has received more attention. The increasing importance in mathematics and signal analysis is due to the link between Gabor superframes generated by Hermite functions which are intrinsically related to polyanalytic spaces [Abr10,GL09]. In [AG10] the theory of Bargmann-Fock spaces has been extended to the setting of polyanalytic functions, see also [AF14] for a survey on these recent developments. One of our main results is a lifting theorem for modulation spaces of Gröchenig-Toft [GT11,GT13] to polyanalytic Bargmann-Fock spaces. Motivated by applications in analytic microlocal analysis and semiclassical quantum theory, in this paper we formulate all results in a semiclassical scaling by assuming that 1 ≫ > 0 is a small parameter.
This paper is structured as follows: after reviewing some basics about Bargmann transforms and quantization in §2, in §3 we introduce the idea of polyanalytic Bargmann transforms as well as polyanalytic Toeplitz quantization T k (m) of a symbol m : C d → C, where k ∈ N d indicates the degree of polyanalyticity. §4 contains our first main theorem, namely, isomorphism results of the form for polyanalytic Toeplitz operators as maps between polyanalytic Sobolev-Fock spaces F k,p,q m (C d ). These spaces appear as images of the well-known modulation spaces under the polyanalytic Bargmann transform. In §5 we present andependent asymptotic symbol calculus for localization operators op ϕ k aw (a), where the window ϕ k is a Hermite function, as well as for their complex counterparts, namely, polyanalytic Toeplitz operators. For example, we show that the commutator of two Hermite localization operators op ϕ k aw (a) and op ϕ k aw (b) has an asymptotic expansion of the form i [ op ϕ k aw (a), op ϕ k aw (b) ] = op ϕ k aw ({a, b}) + O( ) with {·, ·} the usual Poisson bracket on R 2d , and thus corresponds to a O( ) deformation of the classical phase space. Finally, in §6 we apply the new concepts to prove an asymptotic expansion of complex Weyl quantized operators in terms of polyanalytic Toeplitz operators.
In summary, we obtain a whole range of new and related quantization schemes whose combination allows for a refined analysis and more precise approximations. It is the hope of the authors that polyanalytic Toeplitz operators will prove useful in various applications such as manipulation of multiplexed signals, construction and analysis of Gabor superframes and semiclassical quantum dynamics.

Background
We start by reviewing some concepts and results that form the basis for the subsequent introduction and investigation of polyanalytic Toeplitz operators. We first recall Bargmann transforms as well as the well-known Toeplitz, Weyl and anti-Wick quantization schemes. Moreover, for the reader's convenience and later reference we recall the spectrogram expansion of Wigner functions from [Kel19].
2.1. Bargmann transform. The Bargmann transform B -see, e.g., the standard reference [Fol89, §I.6] -maps the usual Hilbert space L 2 (R d ) of quantum mechanics and signal analysis into the Fock space which is a closed subspace of the weighted Hilbert space with the strictly plurisubharmonic exponential weight function Φ(z) = 1 4 |z| 2 and the norm F 2 Hence, the Fock space F (C d ) consists of entire functions of d variables with controlled growth behaviour at infinity. Analoguously to [Abr10], we define the ddimensional -rescaled Bargmann transform as with > 0 a small parameter. In the analytic microlocal setup from [Zwo12,§13], the operator B corresponds to the Fourier-Bros-Iagolnitzer transform associated with the holomorphic quadratic phase (1) θ(z, w) = i 2 ((z − w) 2 − z 2 /2) and the corresponding strictly plurisubharmonic exponential weight function from above that gives rise to the Hilbert space is unitary and the associated orthogonal Bergman projector One computes its adjoint operator B * explicitely as for any function F ∈ L 2 Φ (C d ). Let us consider the image of a Hermite function ϕ k under the Bargmann transform. Hermite functions appear as the eigenfunctions {ϕ k } k∈N d ⊂ L 2 (R d ) of the harmonic oscillator − 2 2 ∆ q + 1 2 |q| 2 , q ∈ R d , and one can show that is an analytic monomial, e.g. by invoking the more general formula in [LT14,Proposition 5]. In particular, Bϕ k is normalized and is an orthonormal basis for F (C d ) consisting of monomials. This property is characteristic to Hermite functions, see e.g. [Jan05]. The Fock space F (C d ) is a reproducing kernel Hilbert space, and the reproducing kernel can be computed explicitly via the Hermite monomial basis (3) as That is, for all z ∈ C d and F ∈ F (C d ) one has the pointwise evaluation property and, as a consequence, one obtains the derivative formula The Bargmann transform B can be seen as the complex equivalent of a specific short-time Fourier transform, which for a general window function u ∈ S(R d ) is defined as with (q, p) ∈ R 2d and the standard translation and modulation operators Namely, for the case of a Gaussian window g 0 := ϕ 0 centered in the origin one observes 2.2. Toeplitz, Weyl and anti-Wick operators. Let us recall the definitions and basic properties of three quantization schemes: Toeplitz, Weyl, and anti-Wick quantization.
The Toeplitz operator T (m) with symbol m : C d → C is defined by multiplication with m and subsequent projection down to the Fock space F (C d ): or, more explicitly, For m ∈ L ∞ (C d ), the quantized operator T (m) is bounded on the Fock space F (C d ). For more general mapping results we refer to §4.1. Weyl quantization or canonical quantization appears as the natural quantization scheme connecting classical and quantum mechanics. Here, a function a : R 2d → C is associated with the Weyl quantized operator op(a) via a( 1 2 (y + q), p)e i(q−y)p/ ψ(y)dy dp where R 2d ∼ = T * R d is the phase space of classical mechanics. The associated phase space representation of quantum states (or signals) is provided by cross-Wigner functions That is, for suitable a, ψ and φ, one has where we choose the inner product to be left-linear. We note that W(ψ, φ) ∈ L 2 (R 2d ) whenever ψ, φ ∈ L 2 (R d ). In the case ψ = φ we write W(ψ, ψ) =: W ψ for the Wigner function to abbreviate notation. Despite of their many remarkable properties, Wigner functions W ψ exhibit the drawback of attaining negative values whenever ψ is not a Gaussian, see [SC83,Jan97], and hence typically are not probability densities. However, one can turn W ψ into a nonnegative function by convolution with another Wigner function: For all ψ ∈ L 2 (R d ) and Schwartz class windows φ ∈ S(R d ) with ψ L 2 (R d ) = φ L 2 (R d ) = 1 the convolution S φ ψ := W ψ * W φ : R 2d → R is a smooth probability density on phase space, as can be deduced from [Fol89, Proposition 1.42]. In time-frequency analysis S φ ψ is called a spectrogram of ψ; see, e.g., the introduction in [Fla13]. Spectrograms constitute a subset of Cohen's class of phase space distributions; see [Fla99, §3.2.1].
A popular window function for spectrograms is provided by the Gaussian wave packet or coherent state , centered in (q, p). We denote the Gaussian wave packet centered in the origin (0, 0) by g 0 . The corresponding spectrogram is known as the Husimi function of ψ, first introduced in [Hus40]. Note that In time-frequency analysis, general anti-Wick type operators op ϕ aw (a), (usually) with a Schwartz class window ϕ, are known as localization operators. Here, they are equivalenty defined via multiplication in the image space of the corresponding short-time Fourier transform (7), op ϕ aw (a) = V * ϕ aV ϕ , op g0 aw (a) = op aw (a), where a denotes both the symbol and the multiplication operator. The non-negative phase space density corresponding to this quantization scheme then in turn is given by the spectrogram S ϕ ψ , see [BCG04]. 2.3. The spectrogram expansion. In the past decades there has been considerable research on the connection between different quantization schemes and their respective calculi, such as the classic comparisons of left, right and Weyl quantization as well as anti-Wick operators, see e.g. [Ler10, §2.3 and §2.4] or [Zwo12, §4 and §13] for summaries.
Explicit formulas for the Wigner and Husimi functions of general wave packets have been derived in [LT14] and subsequently applied in [KLO16] in order to derive second order corrections in the comparison of Wigner and Husimi functions. In [Kel19] these corrections have been generalized to arbitrary order by proving the following spectrogram expansion.
Theorem 1 (Spectrogram expansion from [Kel19]). Let ψ ∈ L 2 (R d ), N ∈ N, and > 0. Then, if one defines the following real-valued phase space function µ N ψ in terms of Hermite spectrograms, for any Schwartz function a : R 2d → C there is a constant C ≥ 0 such that where C only depends on bounds on derivatives of a of degree 2N and higher. In particular, if a is a polynomial with deg(a) < 2N then (18) vanishes.
Retracing the proof for Theorem 1 in [Kel19] immediately shows that the offdiagonal version of the above approximation holds as well. That is, with the offdiagonal phase space representation of any two functions ψ, φ ∈ L 2 (R d ). We note, however, that µ N (ψ, φ) typically has a non-constant complex phase and, in particular, is not a finite linear combination of probability densities. In §6, polyanalytic Toeplitz operators are applied to prove a statement equivalent to Theorem 1 in polyanalytic Bargmann-Fock spaces. This yields a variety of new connections between real and complex Weyl, anti-Wick and Toeplitz type quantization schemes.

Polyanalytic Toeplitz operators
In this section, we first recall the definition of polyanalytic Fock-Bargmann spaces and subsequently introduce and investigate polyanalytic Toeplitz operators which naturally act on these spaces.

Polyanalytic Bargmann-Fock spaces.
Recall that every polyanalytic function F of order k ∈ N d can be uniquely written as where f ℓ , ℓ ∈ N d , are analytic functions and the sum runs over all multiindices with Φ < ∞ the polyanalytic Bargmann-Fock space of degree k which, as we will detail later, has an orthogonal decomposition into true polyanalytic Bargmann-Fock spaces. Note, that polyanalytic functions satisfy a generalized Cauchy-Riemann equation of the form For later reference let us define "translations" in Bargmann-Fock spaces by By once again closely following [Abr10], we then define polyanalytic Bargmann transforms as follows.
in analogy to the definition of Hermite polynomials via their generating function.
As a next step, let us compare B k with the short-time Fourier transform associated with the k-th Hermite function as window just as the zero'th order comparison (8).
Lemma 1 (see e.g. [Abr10]). For all k ∈ N d it holds Proof. By utilizing the partial isometry property of the Bargmann transform and recalling the translation formula (21), for z = q + ip we compute which by means of the differentiation formula (6) leads to the desired result with standard multiindex notation. Since Bf (ℓ) is analytic for all ℓ ∈ N d , B k f is polyanalytic of degree k and the partial isometry property of the polyanalytic Bargmann transforms B k follows directly from the corresponding property of the STFT.
Note that Hermite functions can be used to construct orthonormal bases for polyanalytic function spaces. Namely, the set of transformed Hermite functions and analogously for m ≤ ℓ is an orthonormal basis of denote the Laguerre polynomials, see e.g. [Abr10]. Formula (22) can be proven by using the Laguerre connection for overlap integrals of two shifted Hermite functions similar as for the computation of Wigner transforms of Hermite functions, see e.g. [LT14]. The polynomials in (22) are particular examples of so-called special Hermite functions, see also [RT09].
The polyanalytic Bargmann-Fock spaces admit a decomposition in terms of true polyanalytic Bargmann-Fock spaces namely as the orthogonal sum In particular, recalling (22) we know that for all m ∈ N d the basis function B ℓ ϕ m is a polynomial of degree ℓ in z which implies that all nonzero elements of F k (C d ) share this property as well.
The polyanalytic Bargmann transform B k acts as an isometric isomorphism and, hence, is also known as true polyanalytic Bargmann transform of degree k. In analogy to (2), the map , P = P 0 is the polyanalytic Bergman projector and its kernel the polyanalytic Bergman kernel. The reproducing kernel of F k (C d ) is given by where L k denotes the kth Laguerre polynomial. We furthermore use the notion of polyanalytic Bargmann-Fock spaces of total degree n ∈ N, and the corresponding Bargmann transforms on the span of the true polyanalytic functions of total degree n ∈ N that is related to the Bargmann transform for vector-valued signals from [Abr10] with applications to multiplexing. We denote the corresponding polyanalytic Bergman projector of total degree n ∈ N by P n := B n B * n , n ∈ N. Note that one has the following property: Lemma 2. The polyanalytic Bergman projector of total degree n ∈ N satisfies This follows since in general for the mixed terms it holds B k B * ℓ = 0 though for k = ℓ one still has the orthogonality property B * k B ℓ = 0. 3.2. Polyanalytic Toeplitz quantization. Recall from (16) that general anti-Wick or localization operators are given by where a here denotes both the phase space function a and the operator of multiplication with a. Expectation values of anti-Wick operators are computed on the phase space via the corresponding spectrogram: In the following, we extend the concept of Toeplitz operators as, e.g., defined in [Zwo12, §13] from (9) to the d-dimensional polyanalytic setting, see also [Fau11] for discussions in the one-dimensional case. For defining the quantization, we utilize the polyanalytic Bergman projectors defined in §3.1.
Definition 2 (Polyanalytic Toeplitz quantization). Let k ∈ N d , n ∈ N and f ∈ L ∞ (C d ). Then, the bounded operator is called the true polyanalytic Toeplitz quantization of degree k and true polyanalytic Toeplitz quantization of total degree n.
For the quantization of more general symbols f one needs to introduce corresponding Sobolev type subspaces of F k (C d ) with stronger decay conditions, as we discuss in §4.1.
Note that the Bergman projector on the right-hand side of the multiplication operator in Definition 2 can be safely ommited when acting on polyanalytic Bargmann-Fock spaces. It is included in order to support the intuition that real-valued symbols f ∈ L ∞ (C d , R) give rise to self-adjoint operators.
For later reference, we also define an off-diagonal type polyanalytic Toeplitz quantization by multiplication in the polyanalytic space F k (C d ) and projection back to the usual Fock space F (C d ).
Definition 3 (Projected polyanalytic Toeplitz quantization). Let k ∈ N d and f ∈ L ∞ (C d ). Then, the bounded operator Polyanalytic Toeplitz operators and anti-Wick quantization are closely related in the following way: where we define For later purposes, let us also define the "inverse action" of this map as Relation (23) supports the intuition that localization quantization (16) with Hermite function windows can be seen as the real-valued equivalent of polyanalytic Toeplitz quantization, see also [Fau11].

Polyanalytic Sobolev-Fock spaces and isomorphism theorems
In this section, we first provide a short overview on Sobolev-Fock and modulation spaces that serve as a general class of spaces with natural mapping properties for Toeplitz and localization operators, respectively. Afterwards, we present the polyanalytic generalizations of those spaces and, as a main result, prove an isomorphism theorem for polyanalytic Toeplitz operators. 4.1. Modulation spaces and Sobolev-Fock spaces. Let us briefly review modulation spaces and and their images under the Bargmann transform, the so-called Sobolev-Fock spaces. Modulation spaces form a natural framework for the calculus of localization operators in the same way as Sobolev-Fock spaces do for Toeplitz operators.
Following usual conventions, see e.g. [GT13], we call a locally bounded weight function m : As a result, v is a submultiplicative function and m satisfies We restrict ourselves to weights of polynomial growth and call a weight function admissable if it is moderate, continuous and at most of polynomial growth. For any fixed submultiplicative weight function v : R 2d → (0, ∞) we define the set of v-admissable weights as Then, the modulation spaces with admissible weight m are defined as 1 ≤ p, q ≤ ∞, and contain functions (or distributions) that show controlled growth properties together with their Fourier transforms. We note that modulation spaces do not change if we replace the Gaussian window g 0 by a different Schwartz function, see e.g. [Grö01,§11]. Similarly as the classical Fock space F (C d ) is the image of L 2 (R d ) under the Bargmann transform, one can look at Fock-type spaces that are the equivalents of modulation spaces in the complex setting. We use the notation from [GT13] and write M C v for complex v-admissable weights with v : C d → (0, ∞) moderate. We introduce for any complex moderate weight m the Sobolev-Fock spaces that are complete subspaces of the Banach spaces L p,q Φ,m with the weighted mixed p, q-norm consisting of entire functions. In particular, F 2,2 1 (C d ) = F (C d ) gives the usual Fock space. It is well-known, see e.g. [GT11,GT13], that the Bargmann transform B maps the modulation space M p,q m (R d ) isometrically to the Sobolev-Fock space F q,p m (C d ), where we employ the notation from (24). In particular, from [GT11, Theorem 5.4] we are able to rephrase the following result.
Lemma 3. Let µ ∈ M C w and m ∈ M C v . Then, for all 1 ≤ q, p ≤ ∞, the Toeplitz operator T (m) is a bounded, invertible map from F p,q µ (C d ) to F p,q µ/m (C d ).

4.2.
Polyanalytic Sobolev-Fock spaces. Based on the analytic Sobolev-Fock space theory suitable for Toeplitz operators from §4.1 one can define similar function spaces in the polyanalytic setting. For any k ∈ N d we closely follow the definitions in [AG10] and define true polyanalytic Sobolev-Fock spaces with mixed p, q-norms as As we summarize in the following Lemma 4, polyanalytic Fock-Sobolev spaces are precisely the image of the usual modulation spaces under the polyanalytic Bargmann transform.
Lemma 4. For all 1 ≤ p, q ≤ ∞, k ∈ N d and m ∈ M v , the polyanalytic Bargmann transform B k is an isomorphism Proof. For k = 0 this result is well-known, see e.g. [GT11,GT13,Grö01]. For k = 0 the results follow from Lemma 1 by observing that the modulation space M p,q m (R d ) can be defined without harm with the Hermite window ϕ k instead of g.
Remark 1. We note that -as we stick to weight functions of polynomial growth -the Schwartz space is contained in all considered modulation spaces. This in particular implies that the span of special Hermite functions is a dense subset of F n,p,q m (C d ), see also [RT09]. Moreover, the basis functions B ℓ ϕ m are orthogonal if m is radial in each component, that is, m(z 1 , . . . , z d ) = m(|z 1 |, . . . , |z d |) for somem, see also [GT13]. 4.3. Isomorphism results. In the following, we generalize the isomorphism result from Lemma 3 to the polyanalytic context. For this purpose, we investigate the mapping properties of polyanalytic Toeplitz operators on their respective Sobolev-Fock spaces. This constitutes a main result of this paper.
Theorem 2. Let 1 ≤ p, q ≤ ∞, k ∈ N d , µ ∈ M C w and m ∈ M C v be continuous. Then the polyanalytic Toeplitz operator T k (m) constitutes an isomorphism as a map T k (m) : F k,p,q µ (C d ) → F k,p,q µ/m (C d ) and the k-projected polyanalytic Toeplitz operator T k,0 (m) is an isomorphism Hence, T k (m) can be written as composition of three isomorphisms which completes the proof for the first part of the assertion. For the second part one similarly obtains the diagram for showing the isomorphism property.

Symbol calculus
After we presented the basic concept of polyanalytic Toeplitz operators and their natural action on polyanalytic Sobolev-Fock spaces in the previous sections, we now turn towards a basic symbolic operator calculus for Hermite localization operators as well as polyanalytic Toeplitz operators by providing expansions for compositions and commutators.
For localization operators with symbols in modulation spaces, composition formulas and Fredholm properties have been derived in great generality in [CG06]. Our aim is to obtain more explicit expressions and expansions for small . We start by presenting asymptotic expansions of localization operators with Hermite windows and their compositions as → 0, before moving on to polyanalytic spaces and operators. Lemma 5. Let a : R 2d → C be a Schwartz function, > 0, N ∈ N. Then, with a family r of Schwartz functions satisfying sup >0 op(r ) L 2 →L 2 < ∞.
Let us generalize this formula for higher order Hermite functions. We do this by similar means as applied in [Kel19] for deriving the expansion with Hermite spectrograms. For this purpose, let us recall the formula for Wigner transforms of Hermite functions, where z = (q, p) ∈ R 2d , z j = (q j , p j ) ∈ R 2 and is the kth Laguerre polynomial, see, e.g., [Fol89,§1.9]. In order to generalize Lemma 5 to arbitrary Hermite function windows we have to first get a better understanding of higher order moments of the Wigner transforms of Hermite functions. Note, that due to the symmetry of W ϕ k only even moments are different from zero.
For the proof of Proposition 1, which is mainly built on relations of binomial sums and Gamma functions, we refer to Appendix A. Now, we are ready to generalize Lemma 5 as follows.
Lemma 6. Let k, N ∈ N d , > 0 and a being a Schwartz function. Then, with a family r k of functions satisfying sup >0 op(r k ) L 2 →L 2 < ∞ and the order 2m phase space differential operator D m given by that is a sum of total order 2m differential operators with constant coefficients 2 F 1 (α j + α j+d + 1, −k j ; 1; 2).
Proof. We can basically retrace the idea of [KL13, Lemma 1] by writing and using a Taylor expansion of a around z, Since the symmetry of (26) implies that whenever f is an odd function, the derivatives of odd degree in the Taylor expansion of a do not contribute to the integral (27). For the even degree polynomials we apply Proposition 1 and compute by utilizing the fact that the Wigner function factorizes in the form (26). Hence, which completes the proof as the Calderon-Vaillancourt theorem implies the uniform boundedness of r k .
We would like to stress that due to the fact that the coefficients c α are varying in α it is not straight-forward to write down an inverse expansion as in general D m D n ∝ D m+n unless in the Husimi case k = 0.

Compositions and commutators of Hermite localization operators.
Recall that the composition of two Weyl quantized operators is a Weyl quantized operator again, with the symbol given by the famous Moyal product ♯ of the two symbols, In contrast, the product of two localization operators typically is not a localization operator again. However, the product can be expanded as a sum of localization operators with a regularizing operator as error term that becomes arbitrary small as → 0, see [CG06]. Based on the expansion from Lemma 6, we obtain the following Weyl composition formula for two localization operators that employs the operator A(∇), , acting on the doubled phase space, where σ denotes the standard symplectic form. Note that A(∇) is the generator of bidifferential operators that define the Moyal product expansion.
Proposition 2 (Composition of localization operators). Let k, N ∈ N d , > 0 and a and b be Schwartz functions. Then, with a family ρ k of Schwartz functions satisfying sup >0 op(ρ k ) L 2 →L 2 < ∞ and the total order 2(n + m + ℓ) bidifferential operators where D n has been defined in Lemma 6 and α ℓ in (30).
Proof. We apply Lemma 6 and the expansion of the Moyal product ♯ to compute where ̺ k and ρ k are families of Schwartz functions giving rise to uniformly bounded operator families.
For illustration purposes, let us look at the general expansion from Proposition 2 in the case of second order errors. We compute and observe that D 1 is a diagonally weighted Laplace operator on R 2d , where (2k + 1, 2k + 1) := (2k 1 + 1, 2k 1 + 1, ..., 2k d + 1) ∈ R 2d . Hence, we obtain the following second order composition formula for two localization operators in terms of a Weyl operator: Lemma 7. Let k ∈ N d , > 0 and a and b be Schwartz functions. Then, . In fact, if we allow for second order error terms, the expansion from Lemma 6 can be approximately inverted via with a family θ k of Schwartz functions satisfying sup >0 op(θ k ) L 2 →L 2 < ∞ and the weighted gradient Proof. By combining Proposition 7 and (31) we obtain where, by the Calderon-Vailloncourt theorem, the the second order terms in have a Schwartz class symbol with the desired boundedness properties. Then, calculating implies the result.
From Theorem 3 we directly infer that the commutator of two localization operators exhibits the same Poisson bracket property as the Moyal bracket for Weyl operators with the difference that the error is of second instead of third order in . Remark 2 (Hermite star products). The Hermite star products ⋆ k can be formally defined as of formal power series in with smooth coefficients. Corollary 1 illustrates that -just as the Moyal product ♯-all Hermite star products ⋆ k are compatible with the canonical Poisson structure on phase space. Moreover, the expansion from Lemma 6 implies that the differential star products ⋆ k and ♯ are equivalent for all k in the sense of deformation quantization, see, e.g., [Kon03,BRW07,Sch18]. In particular, we note that the bidifferential operator iα 1 (a, b)+ 1 2 ∇ (k) a, ∇ (k) b from Theorem 3 defines the same 2-cocycle as the Moyal bidifferential operator iα 1 (a, b) in the Hochschild cochain complex over and only differs by the symmetrical coboundary term.
Remark 3 (Anti-Wick star product). In the Husimi case k = 0 one has D n = ∆ n and can explicitly derive higher order versions of Theorem 3. In analogy to the Moyal expansion a simple but tedious calculation yields with the bidifferential operators see [Kel12]. In other words, the operator B(∇) = A(∇) + 1 2 ∇ z , ∇ w generates the bidifferential operators β n that charaterize the anti-Wick star product ⋆ 0 .
The symmetric term ∇ (k) a, ∇ (k) b creates coboundary terms in the bidifferential operators defining Hermite star products and implies that the O( 2 ) error for the commutator expansion in Corollary 1 in general is sharp. In contrast, for the Moyal case the antisymmetry of A(∇) causes O( 3 ) errors for the commutator which is the main ingredient for the the well-known Egorov theorem that allows to link quantum and quasi-classical dynamics with O( 2 ) errors, see [BR02,LR10].
We conclude this section by stressing again that the Weyl operator error term op(θ k ) in Theorem 3 is in general not a localization operator itself. This makes the composition formula purely asymptotic in nature.

5.3.
Calculus of polyanalytic Toeplitz operators. The formulas from §5.2 also imply composition rules for polyanalytic Toeplitz operators as they appear as the complex equivalents of corresponding localization operators with Hermite function windows.
From (23) we first recall the translation formulas between localization operators acting on real-valued signals and polyanalytic Toeplitz operators acting in the complex domain. Moreover, for convenience we introduce the operators w f (z, w), z, w ∈ R 2d , and their complex counterpart where ∂ z , ∂ z as usual denote complex Wirtinger differentials and the diagonal matrix d k = diag(2k, 2k) ∈ R 2d×2d . Note that for k = 0 the second term vanishes and we obtain the simple expression The operator Ξ (k) (∂, ∂) represents Ξ (k) (∇) in the complex domain in the following way.
Lemma 8. Let m, µ : C d → C be smooth and k ∈ N d . Then, it holds where z, w ∈ C d and z R = (Rez, Imz), w R = (Rew, Imw) ∈ R 2d .
With this notation in place, we arrive at the following composition and commutator formulas for polyanalytic Toeplitz operators.
Theorem 4. Let k ∈ N d , > 0 and m, µ : C d → C be Schwartz class functions. Then, it holds with a family θ k of Schwartz functions satisfying sup >0 op(θ k ) L 2 →L 2 < ∞.
Proof. We calculate ( 2 ) by using (32) and applying Theorem 3. From Lemma 8 we then obtain which completes the proof.
Let us remark here, that in the usual Toeplitz quantization case k = 0 this composition formula beautifully reduces to Ignoring all growth restrictions, this shows that whenever µ is analytic (or m antianalytic) the product mµ is the appropriate Toeplitz symbol of the composition up to second order errors.

Weyl quantization and polyanalytic Toeplitz operators
Let us revisit the spectrogram expansion from Theorem 1 with the objects we have defined and investigated so far. By recalling the phase space integral formulas (12) and (15) we first observe that Theorem 1 in fact can be read as a weak approximation of Weyl operators in terms of localization operators with Hermite function windows. In this section, we derive expansions of complex Weyl operators in terms of Bargmann quantized operators and, thus, prove a complex version of Theorem 1. 6.1. An anti-Wick expansion of Weyl operators. By employing the off-diagonal convolution formula and the definition (16) of localization operators we can rewrite (19) as Hence, the spectrogram approximation from Theorem 1 can be rewritten in the following operator form: in the operator norm topology on L 2 (R d ).
One can generalize Proposition 3 in the usual sense by allowing for more general symbol classes. In particular, Proposition 3 remains true as long as a belongs to a suitable Shubin class Γ with z = (1 + |z| 2 ) 1/2 . Note that the Weyl quantization of a symbol a ∈ Γ m ρ (R 2d ) creates a bounded operator from the Shubin-Sobolev space into L 2 (R d ), and it is known that Q m (R d ) actually coincides (with equivalent norm) with the modulation space M 2 z m (R d ), see [BCG04,LR11]. For example, a more general version of Proposition 3 can be formulated as follows.
6.2. Polyanalytic Bargmann representation of antiholomorphic Weyl quantized polynomials. The close connection between polyanalytic Bargmann transforms and short-time Fourier transforms with Hermite windows allows to rephrase Proposition 3 in a Fock space setting. In particular, the important property of almost-invariance of polyanalytic Fock spaces under multiplication with holomorphic polynomials allows to prove the following result that might allow new insights about the manipulation of signals in a multiplexing setup, see [AG10,Abr10].
Proposition 4 (Polyanalytic Bargmann representation of antiholomorphic Weyl operators). Let N ∈ N, > 0 and p : C d → C be a (holomorphic) polynomial of degree N − 1. Then, one has Proof. We start by rewriting the generalized anti-Wick operators in the anti-Wick expansion from Proposition 3 in terms of polyanalytic Bargmann transforms, where the error vanishes because p is of sufficiently low degree, see Theorem 1.
Since p is a holomorphic polynomial, for each true polyanalytic Fock space F k (C d ) multiplication by p leaves a dense subset of F k (C d ) consisting of true polyanalytic polynomials invariant. Moreover, true polyanalytic Fock spaces are othogonal: for any u ∈ F k (C d ) and v ∈ F ℓ (C d ) with k = ℓ it holds Thus, the result follows from observing We can revisit this result in the context of multiplexing as e.g. considered in [AG10]. Namely, polyanalytic Bargmann transforms allow to transform n signals (ψ 0 , . . . , ψ n−1 ) into the single signal that now can jointly be transmitted or manipulated. Afterwards, the n original signals can be recovered via orthogonal projection by using the suitable (polyanalytic) Bergman projectors. This is an implication of the orthogonality of polyanalytic Fock spaces of different degree.
In other words, Proposition 4 can be understood in the sense that the polynomial manipulation of a single multiplexed signal with arbitrary number of "multiplexing copies" can be expressed in terms of the action of usual Weyl operators. For more general manipulations the error terms from the spectrogram expansion can be used when approximating the multi-level Bargmann multiplier by a Weyl operator. 6.3. A polyanalytic Toeplitz expansion of complex Weyl operators. The aim of this section is to provide a version of the anti-Wick expansion from Proposition 3 in the complex setting. That is, instead of anti-Wick operators we employ the earlier defined polyanalytic Toeplitz operators and relate them to complex Weyl operators as considered in [Zwo12,§13].
Let us recall the holomorphic quadratic phase θ from (1) that charaterizes the Bargmann transform. In fact, θ gives rise to the complex symplectic map ) by means of the implicit generating function type definition κ(w, −∂ w θ(z, w)) = (z, ∂ z θ(z, w)), z, w ∈ C d .
One can show that κ is a bijection as a map from R 2d on the Lagrangian subspace The subspace Λ is Im-Lagrangian and Re-symplectic with respect to the complex symplectic form σ C = d j=1 dw ∧ dz on C n × C n , that is, Imσ C ↾ Λ = 0 and Reσ C ↾ Λ is nondegenerate.
In particular Λ is only R-linear but not C-linear and, hence, is not of the type of complex Lagrangian subspaces usually considered in the parametrization of generalized coherent states, see, e.g., [DKT17]. In other words, Λ is an isotropic subspace of maximal dimension, but the Hermitian form is neither positive nor negative definite on Λ, since one computes z, Ωz C 2d = 1 2 (Im(ζ) 2 − Re(ζ) 2 ) ∀z = (ζ, − i 2 ζ) ∈ Λ. In [Zwo12, §13] the symplectic mapping κ from (35) is used to introduce a complex Weyl quantization on the Bargmann transform side. Namely, the bijection κ can be used to identify C d with the Lagrangian subspace Λ ⊂ C 2d and for a Schwartz function a : Λ → C we define its Weyl quantization One can check that op Φ (a) defines a bounded operator both on L 2 Φ (C d ) and the Fock space F (C d ). Now, the Bargmann transform appears as the appropriate translation between real and complex Weyl quantization.
Note that Lemma 9 naturally extends to larger symbol classes, in particular to Shubin classes Γ m ρ (Λ) that consist of functions a for which κ ⋆ a ∈ Γ m ρ (R 2d ), see also (34). We apply Lemma 9 to obtain an expansion of complex Weyl quantized operators in terms of k, 0-polyanalytic Toeplitz operators and, thus, provide a complex version of Proposition 3.
Theorem 5. Let N ∈ N, > 0 and assume a ∈ Γ m ρ (Λ) for m ∈ R, ρ ≥ 0. Then, one has the approximation where κ ⋆ a(z) = κ ⋆ a(q, −p) with z = q + ip ∈ C d . The appropriate decay of op Φ (a)u can be inferred from the intertwining property in Lemma 9 and the maping properties of usual Weyl quantized operators on modulation spaces. Finally, the approximation order follows from Corollary 2.

Outlook
The concept of polyanalytic Toeplitz operators we propose in this paper appears quite straight-forward once written down and naturally exhibits all the favorable mapping qualities that are known from the analytic Bargmann setting. However, by the connection to short-time Fourier transforms and, via the spectrogram expansion, to Weyl operators this new concept allows to formulate profound transition and approximation formulas for the whole range of real and complex Weyl, Toeplitz as well as localization operators.
We believe that polyanalytic Toeplitz quantization might prove a useful concept in a variety of areas, including the deeper investigation and approximation of multiplexed signals, the analysis and generalization of complex quantization theories and a geometrically satisfying complex generalization of coherent state approximations and dynamics.

Appendix A. Moments of special Hermite functions
The Wigner transforms W ϕ k of Hermite functions are also known as special Hermite functions, see, e.g., [Tha93]. Moments of these functions are of special interest as they resemble the quantum expectation values of quantized monomials in the kth harmonic oscillator eigenstate. That is, R 2d z α W ϕ k (z)dz = ϕ k , op(z α )ϕ k with standard multiindex notation, where α ∈ N 2d . As the Wignerfunctions of mulitdimensional Hermite functions factorize into 2-dimensional Wigner functions, in the following we only compute formulas for this case by proving Proposition 1.