The Range of Localization Operators and Lifting Theorems for Modulation and Bargmann-Fock Spaces

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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 365, Number 8, August 2013, Pages 4475–4496 S 0002-9947(2013)05836-9 Article electronically published on January 4, 2013

THE RANGE OF LOCALIZATION OPERATORS AND LIFTING THEOREMS FOR MODULATION AND BARGMANN-FOCK SPACES

KARLHEINZ GROCHENIG¨ AND JOACHIM TOFT

Abstract. We study the range of time-frequency localization operators acting on modulation spaces and prove a lifting theorem. As an application we also characterize the range of Gabor multipliers, and, in the realm of complex analysis, we characterize the range of certain Toeplitz operators on weighted Bargmann-Fock spaces. The main tools are the construction of canonical isomorphisms between modulation spaces of Hilbert-type and a reﬁned version of the spectral invariance of pseudodiﬀerential operators. On the technical level we prove a new class of inequalities for weighted gamma functions.

1. Introduction The precise description of the range of a linear operator is usually difficult, if not impossible, because this amounts to a characterization of which operator equations are solvable. In this paper we study the range of an important class of pseudodifferential operators, so-called time-frequency localization operators, and we prove an isomorphism theorem between modulation spaces with respect to different weights. The guiding example to develop an intuition for our results is the class of multiplication operators. Let m ≥ 0 be a weight function on Rd and define the weighted 1/p p d p p R p | | p space Lm( ) by the norm f Lm = Rd f(x) m(x) dx = fm L .Let M M p a be the multiplication operator defined by af = af.ThenLm is precisely the range of the multiplication operator M1/m. We will prove a similar result for time-frequency localization operators between weighted modulation spaces. To set up terminology, let π(z)g(t)=e2πiξ·tg(t − x) denote the time-frequency shift by z =(x, ξ) ∈ R2d acting on a function g on Rd. The corresponding transform is the short-time Fourier transform of a function defined by −2πiξ·t Vgf(z)= f(t)g(t − x) e dt = f,π(z)g. Rd The standard function spaces of time-frequency analysis are the modulation spaces. The modulation space norms measure smoothness in the time-frequency space (phase space in the language of physics) by imposing a norm on the short-time

Received by the editors October 3, 2011 and, in revised form, March 22, 2012. 2010 Mathematics Subject Classification. Primary 46B03, 42B35, 47B35. Key words and phrases. Localization operator, Toeplitz operator, Bargmann-Fock space, modulation space, Sjöstrand class, spectral invariance, Hermite function. The first author was supported in part by the project P2276-N13 of the Austrian Science Fund (FWF).

c 2013 American Mathematical Society Reverts to public domain 28 years from publication 4475

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Fourier transform of a function f. As a special case we mention the modulation p Rd ≤ ≤∞ spaces Mm( )for1 p and a non-negative weight function m.Let

2 h(t)=2d/4e−πt =2d/4e−πt·t,t∈ Rd p Rd denote the (normalized) Gaussian. Then the modulation space Mm( ) is deﬁned by the norm

p p f Mm = Vhf Lm . g The localization operator Am with respect to the “window” g, usually some test function, and the symbol or multiplier m is defined formally by the integral g Amf = m(z)Vgf(z)π(z)gdz. R2d Localization operators constitute an important class of pseudodifferential operators and occur under different names such as Toeplitz operators or anti-Wick operators. They were introduced by Berezin as a form of quantization [3], and are nowadays applied in mathematical signal processing for time-frequency masking of signals and for phase-space localization [12]. An equivalent form occurs in complex analysis as Toeplitz operators on Bargmann-Fock space [8, 4, 5]. In hard analysis they are used to approximate pseudodifferential operators, in some proofs of the sharp G˚arding inequality and the Fefferman-Phong inequality [24, 25, 26, 30], and in PDE [1]. For the analysis of localization operators with time-frequency methods we refer to [10] and the references given there; for a more analytic point of view we recommend [32, 33]. A special case of our main result can be formulated as follows. By an isomorphism between two Banach spaces X and Y we understand a bounded and invertible operator from X onto Y . Theorem 1.1. Let m be a non-negative continuous symbol on R2d satisfying b m(w + z) ≤ ea|w| m(z) for w, z ∈ R2d, 0 ≤ b<1, and assume that m is radial in each time-frequency coordinate. Then for suitable test functions g the local- g p Rd ization operator Am is an isomorphism from the modulation space Mμ( ) onto p Rd ≤ ≤∞ Mμ/m( ) for all 1 p and moderate weights μ. We see that the range of a localization operator exhibits the same behavior as the multiplication operators. For the precise formulation with all assumptions stated we refer to Section 4. The above isomorphism theorem can also be intepreted as a lifting theorem, quite in analogy with the lifting property of Besov spaces [35]. ˙ p,q However, whereas Besov spaces Bs with different smoothness s are isomorphic via Fourier multipliers, the lifting operators between modulation spaces are precisely the localization operators with the weight m. This case is more subtle because localization operators are not closed under composition, quite in contrast to Fourier multipliers. The isomorphism theorem for localization operators stated above is preceded by many contributions, which were already listed in [21]. In particular, in [11] it was shown that for moderate symbol functions with subexponential growth the localization operator is a Fredholm operator, i.e., it differs from an invertible operator only by a finite-rank operator. Here we show that such operators are isomorphisms between the corresponding modulation spaces, even under weaker conditions than

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needed for the Fredholm property. In the companion paper [21] we proved the isomorphism property for weight functions of polynomial type, i.e., (1) c(1 + |z|)−N ≤ m(z) ≤ C((1 + |z|)N . The contribution of Theorem 1.1 is the extension of the class of possible weights. In particular, the isomorphism property also holds for subexponential weight func- b tions of the form m(z)=ea|z| or m(z)=ea|z|/ log(e+|z|) for a>0, 0

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m and the window g). Since V is a composition of three isomorphisms, we then g p p deduce that Am is an isomorphism from Mμ onto Mμ/m. As an application we prove (i) a new isomorphism theorem for so-called Ga- bor multipliers, which are a discrete version of time-frequency localization operators, and (ii) an isomorphism theorem for Toeplitz operators between weighted Bargmann-Fock spaces of entire functions. To formulate this result more explicitly, ≤ ≤∞ F p Cd for a non-negative weight function μ and 1 p ,let μ( )bethespaceof entire functions of d complex variables deﬁned by the norm p | |p p −pπ|z|2/2 ∞ F F p := F (z) μ(z) e dz < , μ Cd

1 Cd Cd and let P be the usual projection from Lloc( )toentirefunctionson .The Toeplitz operator with symbol m acting on a function F is defined to be TmF = P (mF ). Then we show that the Toeplitz operator Tm is an isomorphism from F p Cd F p Cd ≤ ≤∞ μ( ) onto μ/m( ) for every 1 p and every moderate weight μ. The connection between Toeplitz operators on Bargmann-Fock space and localization operators is also used in [13]. The paper is organized as follows: In Section 2 we provide the precise definition of modulation spaces and localization operators, and we collect their basic properties. In particular, we investigate the Weyl symbol of the composition of two localization operators. In Section 3, which contains our main contribution, we construct the canonical isomorphisms and prove Theorem 1.2. In Section 4 we derive a refinement of the spectral invariance of pseudodifferential operators and prove the general isomorphism theorem, of which Theorem 1.1 is a special case. Finally in Section 5 we give applications to Gabor multipliers and Toeplitz operators.

2. Time-frequency analysis and localization operators We ﬁrst set up the vocabulary of time-frequency analysis. For the notation we follow the book [17]. For a point z =(x, ξ) ∈ R2d in phase space the time-frequency shift of a function f is π(z)f(t)=e2πiξ·tf(t − x), t ∈ Rd.

∈ 1 Rd The short-time Fourier transform: Fix a non-zero function g Lloc( ) which is usually taken in a suitable space of Schwartz functions. Then the short- time Fourier transform of a function or distribution f on Rd is deﬁned to be

2d (2) Vgf(z)=f,π(z)g z ∈ R ,

provided the scalar product is well-deﬁned for every z ∈ R2d.Hereg is called a “window function”. If f ∈ Lp(Rd)andg ∈ Lp (Rd) for the conjugate parameter p = p/(p − 1), then the short-time Fourier transform can be written in integral form as −2πiξ·t Vgf(z)= f(t)g(t − x) e dt. Rd In general, the bracket ·, · extends the inner product on L2(Rd) to any dual pairing between a distribution space and its space of test functions, for instance g ∈S(Rd) and f ∈S(Rd), but time-frequency analysis often needs larger distribution spaces.

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Weight functions: We call a continuous, strictly positive weight function m on R2d moderate if m(z + y) m(z − y) sup , := v(y) < ∞ for all y ∈ R2d. z∈R2d m(z) m(z) The resulting function v is a submultiplicative weight function, i.e., v is even and 2d satisfies v(z1 + z2) ≤ v(z1)v(z2) for all z1,z2 ∈ R ,andthenm satisfies 2d (3) m(z1 + z2) ≤ v(z1)m(z2) for all z1,z2 ∈ R . Given a submultiplicative weight function v on R2d, any weight satisfying condition(3)iscalledv-moderate. For a fixed submultiplicative function v the set M { ∈ ∞ R2d ≤ ∀ ∈ R2d} v := m Lloc( ):0

1 m(z1) 2d (4) ≤ ≤ v(z1 − z2) for all z1,z2 ∈ R . v(z1 − z2) m(z2)

This follows from (3) by replacing z1 with z1 − z2.

p,q Modulation spaces Mm for arbitrary weights: For the general definition of modulation spaces we choose the Gaussian function h(t)=2d/4e−πt·t as the canonical window function. Then the short-time Fourier transform is defined for arbitrary 1/2 Rd elements in the Gelfand-Shilov space (S1/2 ) ( ) of generalized functions. The mod- p,q Rd ≤ ∞ ∈ 1/2 Rd ulation space Mm ( ), 1 p, q < consists of all elements f (S1/2 ) ( )such that the norm q/p 1/q p p p,q | | p,q (5) f Mm := Vhf(x, ξ) m(x, ξ) dx dξ = Vhf Lm Rd Rd is finite. If p = ∞ or q = ∞, we make the usual modification and replace the p,q integral by the supremum norm ·L∞ .Ifm = 1, then we usually write M p,q p p,p p p,p instead of Mm .WealsosetMm = Mm and M = M . p,q Rd The reader who does not like general distribution spaces may interprete Mm ( ) as the completion of the finite linear combinations of time-frequency shifts H0 = { ∈ R2d} p,q ≤ ∞ span π(z)h : z with respect to the Mm -norm for 1 p, q < and as a weak∗-relative closure, when p = ∞ or q = ∞. These issues arise only for extremely ≥ ≤ ≤ p,q Rd rapidly decaying weight functions. If m 1and1 p, q 2, then Mm ( )isin 2 Rd p,q Rd fact a subspace of L ( ). If m is of polynomial type (cf. (1)), then Mm ( )is a subspace of tempered distributions. This is the case that is usually considered, although the theory of modulation spaces was developed from the beginning to include arbitrary moderate weight functions [14, 17, 19].

Norm equivalence: Deﬁnition (5) uses the Gauss function as the canonical window. The deﬁnition of modulation spaces, however, does not depend on the ∈ 1 ∈M particular choice of the window. More precisely, if g Mv , g =0,andm v, then there exist constants A, B > 0 such that

p,q ≤ p,q ≤ p,q p,q (6) A f Mm Vgf Lm B f Mm = B Vhf Lm .

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We will usually write

p,q p,q Vgf Lm f Mm for the equivalent norms.

∈ 1 Rd Localization operators: Given a non-zero window function g Mv ( )and R2d g a symbol or multiplier m on , the localization operator Am is defined informally by g (7) Amf = m(z)Vgf(z)π(z)gdz, R2d g provided the integral exists. A useful alternative definition of Am is the weak definition g (8) Amf,k L2(Rd) = mVgf,Vgk L2(R2d). While in general the symbol m may be a distribution in a modulation space of ∞ R2d the form M1/v( ) [10, 33], we will investigate only localization operators whose symbol is a moderate weight function. Taking the short-time Fourier transform of (7), we find that

g Vg(Amf)(w)= m(z)Vgf(z) π(z)g, π(w)g dz = (mVgf) Vgg (w) , R2d with the usual twisted convolution deﬁned by · − (FG)(w)= F (z)G(w − z)e2πiz1 (z2 w2) dz. R2d Since F → F (z)π(z)g, π(·)g dz R2d is the projection from arbitrary tempered distributions on R2d onto functions of the form Vgf for some distribution f, the localization operator can be seen as the composition of a multiplication operator and the projection onto the space of short- time Fourier transforms for the ﬁxed window g. In this light, localization operators resemble the classical Toeplitz operators, which are multiplication operators fol- lowed by a projection onto analytic functions. Therefore they are sometimes called Toeplitz operators [21, 31, 33]. If the window g is chosen to be the Gaussian, then this formal similarity can be made more precise. See Proposition 5.5. 2.1. Mapping properties of localization operators. The mapping properties of localization operators on modulation spaces closely resemble the mapping properties of multiplication operators between weighted Lp-spaces. The boundedness of localization operators has been investigated on many levels of generality [10, 32, 36]. We will use the following boundedness result from [11, 33]. ∈M ∈M ∈ 1 Rd Lemma 2.1. Let m v and μ w.Fixg Mvw( ). Then the localization g p,q Rd p,q Rd operator A1/m is bounded from Mμ ( ) to Mμm( ).

Remark. The condition on the window g is required to make sense of Vgf for f in p,q Rd p ,q Rd p,q the domain space Mμ ( )andofVgk for k in the dual M1/(μm)( )=(Mμm) p,q ∈ ∞ of the target space Mμm for the full range of parameters p, q [1, ]. For ﬁxed p, q ∈ [1, ∞] weaker conditions may suﬃce, because the norm equivalence (6) still ∈ 1 ∈ r ≤ holds after relaxing the condition g Mv into g Mv for r min(p, p ,q,q ) [34].

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g On a special pair of modulation spaces, A1/m is even an isomorphism [21]. ∈ 1 ∈M 1/2 g Lemma 2.2. Let g Mv , m v,andsetθ = m .ThenAm is an isomor- 2 Rd 2 Rd phism from Mθ ( ) onto M1/θ( ). 2 Rd 2 Rd Likewise A1/m is an isomorphism from M1/θ( ) onto Mθ ( ). Consequently g g 2 Rd g g the composition A1/mAm is an isomorphism on Mθ ( ),andAmA1/m is an iso- 2 Rd morphism on M1/θ( ). Lemma 2.2 is based on the equivalence g | |2 · 2 2 (9) Amf,f = m, Vgf = Vgf θ 2 f 2 , Mθ and is proved in detail in [21, Lemma 3.4]. g g 2.2. The symbol of A1/mAm. The composition of localization operators is no longer a localization operator, but the product of two localization operators still has a well-behaved Weyl symbol. In the following we use the time-frequency calculus of pseudodifferential operators as developed in [18, 20]. Compared to the standard pseudodifferential operator calculus it is more restrictive because it is related to the constant Euclidean geometry on phase space. On the other hand, it is more general because it works for arbitrary moderate weight functions (excluding exponential growth). Given a symbol σ(x, ξ)onRd × Rd R2d, the corresponding pseudodifferential operator in the Weyl calculus Op(σ)isdefinedformallyas x + y Op(σ)f(x)= σ ,ξ e2πi(x−y)·ξf(y) dydξ R2d 2 with a suitable interpretation of the integral. If C is a class of symbols, we write Op(C)={Op(σ):σ ∈C}for the class of all pseudodifferential operators with symbols in C. For the control of the symbol of composite operators we will use the following characterization of the generalized Sjöstrand class from [18]. For the formulation associate to a submultiplicative weight v(x, ξ)onR2d the rotated weight on R4d defined by (10)v ˜(x, ξ, η, y)=v(−y, η). For radial weights, to which we will restrict later, the distinction between v andv ˜ is unnecessary. ∈ 1 Theorem 2.3. Fix a non-zero g Mv .AnoperatorT possesses a Weyl symbol ∞,1 ∈ ∞,1 in Mv˜ , T Op(Mv˜ ), if and only if there exists a (semi-continuous) function ∈ 1 R2d H Lv( ) such that |Tπ(z)g, π(y)g| ≤ H(y − z) for all y, z ∈ R2d. ∞,1 Remark. This theorem says the symbol class Mv˜ is characterized by the off- diagonal decay of its kernel with respect to time-frequency shifts. This kernel is in fact dominated by a convolution kernel. The composition of operators can then be studied with the help of convolution relations. Clearly this is significantly easier than the standard approaches that work with the Weyl symbol directly and the twisted product between Weyl symbols. See [22] for results in this direction. ∈ 1 Rd ∈ ∞,1 ≥ Theorem 2.4. Assume that g Mvs ( ), T Op(Mv˜s ) for s 1/2,and ∈M g g ∈ ∞,1 θ v1/2 .ThenAθTA1/θ Op(Mv˜s−1/2 ).

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g Proof. We distinguish the window g of the localization operator Am from the window h used in the expression of the kernel Tπ(z)h, π(y)h.Chooseh to be the ∈ 1 Gaussian; then h Mv for every submultiplicative weight v. Let us ﬁrst write the kernel Tπ(z)h, π(y)h informally and later justify the convergence of the integrals. Recall that g −1 T (A1/θf)=T θ(u) f,π(u)g π(u)gdu . R2d Then g g g g (11) AθTA1/θπ(z)h, π(y)h = TA1/θπ(z)h, Aθπ(y)h 1 = π(z)h, π(u)g Tπ(u)g, π(u)g θ(u) π(y)h, π(u)g dudu . R4d θ(u) Now set ∗ G(z)=|g, π(z)h| = |Vhg(z)| and G (z)=G(−z) 1 R2d and let H be a dominating function in Lv( )sothat |Tπ(u)g, π(u)g| ≤ H(u − u). Since time-frequency shifts commute up to a phase factor, we have |π(z)h, π(u)g| = G(z − u). √ Before substituting all estimates into (11), we recall that θ is v-moderate by assumption, and so (4) says that θ(u) ≤ v(u − u)1/2 for all u, u ∈ R2d. θ(u) Now by (11) we get |AgTAg π(z)h, π(y)h| θ 1/θ θ(u) ≤ G(z − u)H(u − u)G(y − u) dudu R2d R2d θ(u) ≤ G(z − u)v(u − u)1/2H(u − u)G(y − u) dudu R2d R2d = G ∗ (v1/2H) ∗ G∗ (z − y).

g g ∗ 1/2 ∗ ∗ Thus the kernel of AθTA1/θ is dominated by the function G (v H) G .By ∈ 1 Rd ∈ 1 R2d ∈ ∞,1 assumption g Mvs ( ) and thus G Lvs ( ), and T Op(Mv˜s ) and thus ∈ 1 R2d 1/2 ∈ 1 R2d H Lvs ( ). Then v H Lvs−1/2 ( ). Consequently ∗ 1/2 ∗ ∗ ∈ 1 ∗ 1 ∗ 1 ⊆ 1 (12) G (v H) G Lvs Lvs−1/2 Lvs Lvs−1/2 . g g ∈ ∞,1 The characterization of Theorem 2.3 now implies that AθTA1/θ Op(Mv˜s−1/2 ).

∈ 1 Rd ∈M Corollary 2.5. Assume that g Mv2w( ) and m v and w is an arbitrary g g ∈ ∞,1 submultiplicative weight. Then A1/mAm Op(Mv˜w˜ ).

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∈ ∞,1 Proof. In this case T is the identity operator and Id Op(Mv0 ) for every submul- − ∈ ∞,1 tiplicative weight v0(x, ξ, η, y)=v0( y, y). In particular, Id Op(Mv2 ). Now ∈M ∈M replace the weight θ in Theorem 2.4 by m and the condition θ v1/2 by m v and modify the convolution inequality (12) in the proof of Theorem 2.4.

3. Canonical isomorphisms between modulation spaces of Hilbert-type In [21] we have used a deep result of Bony and Chemin [7] regarding the existence of isomorphisms between modulation spaces of Hilbert-type and then extended those isomorphisms to arbitrary modulation spaces. Unfortunately the result of Bony and Chemin is restricted to weights of polynomial-type and does not cover b weights moderated by superfast growing functions, such as v(z)=ea|z| for 0 < b<1. In this section we construct explicit isomorphisms between L2(Rd) and the modu- 2 Rd lation spaces Mθ ( ) for a general class of weights. We will assume that the weights are radial in each time-frequency variable. Precisely, consider time-frequency variables (13) (x, ξ) z = x + iξ ∈ Cd R2d, which we identify by

d 2d (x1,ξ1; x2,ξ2; ...; xd,ξd)=(z1,z2,...,zd) ∈ C R . Then the weight function m should satisfy

2d (14) m(z)=m0(|z1|,...,|zd|)forz ∈ R

Rd ∞ d for some function m0 on + =[0, ) . Without loss of generality, we may also assume that m is continuous on R2d. (Recall that only weights of polynomial-type occur in the lifting results in [21]. On the other hand, no radial symmetry is needed in [21].) ∈ Nd For each multi-index α =(α1,...,αd) 0 we denote the corresponding multi- variate Hermite function by

d 1/4 n/2 n 2 π 2 d 2 h (t)= h (t ), where h (x)= eπx (e−2πx ) α αj j n n!1/2 dxn j=1

is the n-th Hermite function in one variable with the normalization hn2 =1. Then the collection of all Hermite functions hα,α ≥ 0, is an orthonormal basis of L2(Rd). By identifying R2d with Cd via (13), the short-time Fourier transform d/4 −πt2 of hα with respect to h(t)=2 e is simply (15) |α| π 1/2 2 2 V h (z)=e−πix·ξ zα e−π|z| /2 = e−πix·ξ e (z) e−π|z| for z ∈ Cd. h α α! α Remark. We mention that a formal Hermite expansion f = α cαhα deﬁnes a 1/2 distribution in the Gelfand-Shilov space (S1/2 ) if and only if the coeﬃcients satisfy |α| ∗ |cα| = O(e ) for every >0. The Hermite expansion then converges in the weak 1/2 × 1/2 topology. Here we have used the fact that the duality (S1/2 ) S1/2 extends the

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2 · · 1/2 2 Rd × L -form , on S1/2 , and likewise the duality of the modulation spaces Mθ ( ) 2 Rd M1/θ( ). Consequently, the coeﬃcient cα of a Hermite expansion is uniquely ∈ 1/2 determined by cα = f,hα for f (S1/2 ) . See [23] for details. In the following we take for granted the existence and convergence of Hermite expansions for functions and distributions in arbitrary modulation spaces. By 2 dμ(z)=e−π|z| dz we denote the Gaussian measure on Cd. Lemma 3.1. Assume that θ(z)=O(ea|z|) and that θ is radial in each coordinate. α ≥ 2 Cd (a) Then the monomials z ,α 0, are orthogonal in Lθ( ,μ). 2 Rd (b) The ﬁnite linear combinations of the Hermite functions are dense in Mθ ( ).

iϕj By using polar coordinates zj = rj e ,whererj ≥ 0andϕj ∈ [0, 2π), we get α α iα·ϕ (16) z = r e and dz = r1 ···rd dϕdr, and the condition on θ in Lemma 3.1 can be recast as

(17) θ(z)=θ0(r), d for some appropriate function θ0 on [0, ∞) ,andr =(r1,...,rd)andϕ=(ϕ1,...,ϕd) as usual.

Proof. (a) This is well known and is proved in [12, 16]. In order to be self-contained, we recall the arguments. By writing the integral over R2d in polar coordinates in each time-frequency pair, (16) and (17) give 2 2 α β 2 −π|z| i(α−β)·ϕ α+β −πr 2 z z θ(z) e dz = e r e θ0(r) r1 ···rd dϕdr. R2d Rd × d + [0,2π)

The integral over the angles ϕj is zero, unless α = β, whence the orthogonality of the monomials. 2 Rd (b) Density: Assume on the contrary that the closed subspace in Mθ ( ) spanned 2 Rd by the Hermite functions is a proper subspace of Mθ ( ). Then there exists a non- zero f ∈ (M 2(Rd)) = M 2 (Rd) such that f,h = 0 for all Hermite functions θ 1/θ α ∈ 2 Rd ∈ Nd hα Mθ ( ), α 0. Consequently the Hermite expansion of f = α f,hα hα = 1/2 0in(S1/2 ) , which contradicts the assumption that f =0.

h Deﬁnition 1. The anti-Wick operator Jm is the localization operator Am associated to the weight m and to the Gaussian window h = h0. Speciﬁcally,

(18) Jmf = m(z)f,π(z)hπ(z)hdz. R2d For m = θ2 we obtain 2 2 (19) Jmf,f = mVhf,Vhf R2d = Vhfθ 2 = f 2 Mθ whenever f is in a suitable space of test functions. Localization operators with respect to Gaussian windows have been used in sev- eral problems of analysis; see for instance [1, 24, 25, 27]. They have rather special properties. In view of the connection to the localization operators on the Bargmann- Fock space (see below), this is to be expected.

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Theorem 3.2. If θ is a continuous, moderate function and radial in each time- frequency coordinate, then each of the mappings 2 Rd → 2 Rd 2 Rd → 2 Rd Jθ : Mθ ( ) L ( ),Jθ : L ( ) M1/θ( ), 2 Rd → 2 Rd 2 Rd → 2 Rd J1/θ : M1/θ( ) L ( ),J1/θ : L ( ) Mθ ( ) is an isomorphism. The special case of the polynomial weight θ(z)=(1+|z|2)1/2 was already proved in [27] with completely different methods. The extension to weights of non-polynomial growth is non-trivial and requires a number of preliminary results. In these investigations we will play with different coefficients of the form

τα(θ):=Jθhα,hα, or, more generally, |α| s s π α 2 −π|z|2 (20) τα,s(θ):=τα(θ )=Jθs hα,hα = θ(z) |z | e dz , R2d α!

when θ is a weight function and s ∈ R. We note that the τα,s(θ) are strictly positive, since θ is positive. If θ ≡ 1, then τα,s(θ) = 1, so we may consider the coefficients τα,s(θ)α!asweighted gamma functions. 2 Proposition 3.3 (Characterization of Mθ with Hermite functions). Let θ be a moderate and radial function. Then 2 2 2 (21) f 2 = |f,hα| τα(θ ). Mθ α≥0 Proof. Let f = α≥0 cαhα be a finite linear combination of Hermite functions. Since the short-time Fourier transform of f with respect to the Gaussian h is given by −πix·ξ −π|z|2/2 Vhf(z)= cαVhhα(z)=e cαeα(z)e α≥0 in view of (15), definition (18) gives 2 | |2 2 f M 2 = Vhf(z) θ(z) dz θ R2d 2 −π|z|2 2 2 = cαcβ eα(z)eβ(z)θ(z) e dz = |cα| τα(θ ). R2d α,β≥0 α≥0 In the latter equalities it is essential that the weight θ is radial in each time- 2 Cd frequency coordinate so that the monomials eα are orthogonal in Lθ( ,μ). In the next proposition, which is due to Daubechies [12], we represent the anti- Wick operator by a Hermite expansion. Proposition 3.4. Let θ be a moderate, continuous weight function on R2d that is radial in each time-frequency coordinate. Then the Hermite function hα is an eigenfunction of the localization operator Jθ with eigenvalue τ (θ) for α ∈ Nd,andJ possesses the eigenfunction expansion α 0 θ 2 d (22) Jθf = τα(θ)f,hαhα for all f ∈ L (R ). α≥0

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Proof. By Lemma 3.1(a) we ﬁnd that, for α = β,

Jθhβ,hα = θ(z)Vhhβ(z) Vhhα(z) dz Cd −π|z|2 = θ(z)eβ(z)eα(z)e dz =0. Cd This implies that Jθhα = chα, and therefore c = c hα,hα = Jθhα,hα = τα(θ). For a (ﬁnite) linear combination f = β≥0 cβhβ,weobtain Jθf = Jθf,hαhα = cβJθhβ,hαhα α≥0 α≥0 β≥0 = τβ(θ)δα,βcβhα = τα(θ)cαhα. α≥0 β≥0 α≥0

2 Rd The proposition follows because the Hermite functions span M1/θ( ) and because the coeﬃcients of a Hermite expansion are unique and given by cα = f,hα).

2 d Corollary 3.5. If θ is moderate and radial in each coordinate, then Jθ : L (R ) → 2 Rd 2 Rd M1/θ( ) is one-to-one and possesses dense range in M1/θ( ). Proof. The coeﬃcients in Jθf = α≥0 τα(θ) f,hα hα are unique. If Jθf =0,then τα(θ)f,hα = 0, and since τα(θ) > 0weobtainf,hα =0andthusf = 0. Clearly 2 Rd the range of Jθ in M1/θ( ) contains the ﬁnite linear combinations of Hermite 2 Rd functions, and these are dense in M1/θ( ) by Lemma 3.1(b).

2 Rd 2 Rd To show that Jθ maps L ( )ontoM1/θ( ) is much more subtle. For this we need a new type of inequality valid for the weighted gamma functions in (20). By Proposition 3.4 the number τα,s(θ) is exactly the eigenvalue of the localization operator Jθs corresponding to the eigenfunction hα.

Proposition 3.6. If θ ∈Mw is continuous and radial in each time-frequency coordinate, then the mapping s → τα,s(θ) is “almost multiplicative”. This means that for every s, t ∈ R there exists a constant C = C(s, t) such that

−1 (23) C ≤ τα,s(θ)τα,t(θ)τα,−s−t(θ) ≤ C for all multi-indices α.

Proof. The upper bound is easy. By Proposition 3.4 the Hermite function hα is a common eigenfunction of Jθs ,Jθt ,andJθ−s−t . Since the operator Jθs Jθt Jθ−s−t is bounded on L2(Rd) by repeated application of Lemma 2.1, we obtain that

(24) τα,s(θ)τα,t(θ)τα,−s−t(θ)=Jθs Jθt Jθ−s−t hαL2 ≤ ChαL2 = C

2 d for all α ≥ 0. The constant C is the operator norm of Jθs Jθt Jθ−s−t on L (R ). For the lower bound we rewrite the deﬁnition of τα,s(θ) and make it more explicit iϕj by using polar coordinates zj = rj e ,rj ≥ 0,ϕj ∈ [0, 2π), in each variable. Then Rd by assumption θ(z)=θ0(r) for some continuous moderate function θ0 on +,and

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we obtain |α| s π α 2 −π|z|2 (25) τα,s(θ)= θ(z) |z | e dz R2d α! |α| d π 2α −π|r|2 =(2π) θ0(r) r e r1 ···rd dr Rd α! + ∞ ∞ d d 1 2 α −πr2 =(2π) ... θ (r ,...,r ) (πr ) j e j r ···r dr ···dr 0 1 d α ! j 1 d 1 d 0 0 j=1 j ∞ ∞ d αj uj − = ... θ u /π,..., u /π e uj du ···du . 0 1 d α ! 1 d 0 0 j=1 j

n −x We ﬁrst focus on a single factor in the integral. The function fn(x)=x e /n! takes its maximum at x = n,and 1 f (n)= nne−n =(2πn)−1/2 1+O(n−1) n n! − √by Stirling’s√ formula. Furthermore,√ fn is almost constant on the interval [n n/2,n+ n/2] of length√ n. On this interval the minimum of fn is taken at one of the endpoints n ± n/2,wherethevalueis √ n √ √ 1 √ 1 n 1 n f (n ± n/2) = (n ± n/2)ne−(n± n/2) = 1 ± √ e∓ n/2. n n! n! en 2 n Since n 1/2 n (2πn) √ 1 n lim e = 1 and lim 1 ± √ e∓ n/2 = e−1/8 n→∞ n! n→∞ 2 n by Stirling’s formula and straightforward applications of Taylor’s formula, we ﬁnd that c √ √ (26) f (x) ≥ √ for x ∈ [n − n/2,n+ n/2] and all n ≥ 1. n n

−1/2 For n = 0 we use the inequality f0(x) ≥ e for x ∈ [0, 1/2]. Now consider the products of the fn’s occuring in the integral above. For α = ∈ Nd (α1,...,αd) 0 deﬁne the boxes d √ √ −1 −1 d Cα = αj − 2 αj ,αj +2 max( αj , 1) ⊆ R , j=1 d −1 with volume vol (Cα)= j=1 max(2 ,αj ). Consequently, on the box Cα we have

d αj d u 1 −1 j −uj (27) e ≥ C0 = C0 vol Cα α ! −1 j=1 j j=1 max(2 ,αj ) ∈ Nd for some constant C0 > 0 which is independent of α 0.

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Next, to take into account the coordinate change in (25), we deﬁne the box d − √ − √ (α − 2 1 α )1/2 (α +2 1 max( α , 1))1/2 D = j √ j , j √ j ⊆ Rd . α π π j=1

Furthermore, the length of each edge of Dα is √ √ −1/2 1/2 1/2 −1/2 π αj + αj /2 − αj − αj /2 ≤ π

when αj ≥ 1, and likewise for αj =0.Consequently, −1/2 −1/2 d (28) if z1,z2 ∈ Dα, then z1 − z2 ⊆ [−π ,π ] .

After these preparations we start the lower estimate of τα,s(θ). Using (27) we obtain ∞ ∞ d αj s uj − τ (θ)= ... θ u /π,..., u /π e uj du ···du α,s 0 1 d α ! 1 d 0 0 j=1 j 1 s ≥ C0 θ0 u1/π,..., ud/π du1 ···dud. vol (Cα) Cα Since θ is continuous, the mean value theorem asserts that there is a point z = z(α, s)=(z ,z ,...,z ) ∈ C such that 1 2 d α s τα,s(θ) ≥ C0 θ0 z1/π,..., zd/π . Note that the point with coordinates ζ = ζ(α, s)= z1/π,..., zd/π is in Dα; consequently, s τα,s(θ) ≥ C0 θ0(ζ(α, s)) for ζ(α, s) ∈ Dα. Finally ≥ 3 s t −s−t (29) τα,s(θ) τα,t(θ) τα,−s−t(θ) C0 θ0(ζ) θ0(ζ ) θ0(ζ ) for points ζ,ζ ,ζ ∈ Dα. Since the weight θ is a w-moderate, θ0 satisﬁes

θ0(z1) 1 d ≥ ,z1,z2 ∈ R . θ0(z2) w(z1 − z2) −1/2 −1/2 d Since ζ,ζ ,ζ ∈Dα, the diﬀerences ζ−ζ and ζ −ζ are in the cube [−π ,π ] as observed in (28). We conclude the non-trivial part of this estimate by

3 1 1 τ (θ)τ (θ)τ − − (θ) ≥ C α,s α,t α, s t 0 w(ζ − ζ)s w(ζ − ζ)t −s−t ≥ 3 C0 max w(z) = C. z∈[−π−1/2,π−1/2]d The proof is complete. The next result provides a sort of symbolic calculus for the anti-Wick operators Jθs . Although the mapping s → Jθs is not a homomorphism from R to operators, it is multiplicative modulo bounded operators. Theorem 3.7. Let θ and μ be two moderate, continuous weight functions on R2d that are radial in each time-frequency variable. For every r, s ∈ R there exists an 2 Rd operator Vs,t that is invertible on every Mμ( ) such that

Jθs Jθt Jθ−s−t = Vs,t.

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Proof. For s, t ∈ R ﬁxed, set γ(α)=τα,s(θ)τα,t(θ)τα,−s−t(θ)and Vs,tf = γ(α)f,hαhα. α≥0

−1 Clearly, Vs,t = Jθs Jθt Jθ−s−t .SinceC ≤ γ(α) ≤ C for all α ≥ 0byProposi- 2 tion 3.6, Proposition 3.3 implies that Vs,t is bounded on every modulation space Mμ. −1 −1 Likewise the formal inverse operator Vs,t f = α≥0 γ(α) f,hα hα is bounded on 2 Rd 2 Rd Mμ( ); consequently, Vs,t is invertible on Mμ( ).

We can now ﬁnish the proof of Theorem 3.2.

Proof of Theorem 3.2. Choose s =1andt = −1; then JθJ1/θ = V1,−1 is invertible 2 on L . Similarly, the choice s = −1,t =1yieldsthatJ1/θJθ = V−1,1 is invertible 2 2 on Mθ . The factorization JθJ1/θ = V1,−1 implies that J1/θ is one-to-one from L 2 2 2 to Mθ and that Jθ maps Mθ onto L . The factorization J1/θJθ = V−1,1 implies 2 2 2 2 that Jθ is one-to-one from Mθ to L and that J1/θ maps L onto Mθ . 2 2 We have proved that Jθ is an isomorphism from Mθ to L and that J1/θ is an 2 2 isomorphism from L to Mθ . The other isomorphisms are proved similarly.

Remark. In dimension d = 1 the invertibility of JθJθ−1 follows from the equivalence τn,1(θ)τn,−1(θ) 1, which can be expressed as the following inequality for weighted gamma functions: ∞ n ∞ n −1 x −x 1 x −x (30) C ≤ θ0( x/π) e dx e dx ≤ C 0 n! 0 θ0( x/π) n!

for all n ≥ 0. Here θ0 is the same as before. It is a curious and fascinating fact that this inequality implies that the localization operator J1/θ is an isomorphism 2 R 2 R between L ( )andMθ ( ).

4. The general isomorphism theorems In Theorem 4.3 we will state the general isomorphism theorems. The strategy of the proof is similar to that of Theorem 3.2 in [21]. The main tools are the theorems about the spectral invariance of the generalized Sj¨ostrand classes [18] and 2 Rd 2 Rd the existence of a canonical isomorphism between L ( )andMθ ( ) established in Theorem 3.2.

4.1. Variations on spectral invariance. We ﬁrst introduce the tools concerning the spectral invariance of pseudodiﬀerential operators. Recall the following results from [18]. Theorem 4.1. Let v be a submultiplicative weight on R2d such that (31) lim v(nz)1/n =1 for all z ∈ R2d , n→∞ ∈ ∞,1 2 Rd and let v˜ be the same as in (10).IfT Op(Mv˜ ) and T is invertible on L ( ), −1 ∈ ∞,1 then T Op(Mv˜ ). p,q Rd Consequently, T is invertible simultaneously on all modulation spaces Mμ ( ) for 1 ≤ p, q ≤∞and all μ ∈Mv.

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Condition (31) is usually called the Gelfand-Raikov-Shilov (GRS) condition. We prove a more general form of spectral invariance. Since we have formulated all results about the anti-Wick operators Jθ for radial weights only, we will assume from now on that all weights are radial in each coordinate. In this case v˜(x, ξ, η, y)=v(−η, y)=v(y, η), and we do not need the somewhat ugly distinction between v andv ˜.

2 Theorem 4.2. Assume that v satisﬁes the GRS condition, θ ∈Mv, and that both v and θ are radial in each time-frequency coordinate. ∈ ∞,1 2 Rd 2 Rd If T Op(Mv ) and T is invertible on Mθ ( ),thenT is invertible on L ( ). p,q Rd ≤ As a consequence T is invertible on every modulation space Mμ ( ) for 1 p, q ≤∞and μ ∈Mv.

2 d Proof. Set T = JθTJ1/θ. By Theorem 3.2, J1/θ is an isomorphism from L (R ) 2 Rd 2 Rd 2 Rd onto Mθ ( )andJθ is an isomorphism from Mθ ( )ontoL ( ); therefore T is an isomorphism on L2(Rd).

2 −→T 2 Mθ Mθ (32) ↑ J1/θ ↓ Jθ L2(Rd) −→T L2(Rd) ∞,1 2 Rd By Theorem 2.4 the operator T is in Op(Mv1/2 ). Since T is invertible on L ( ), ∞,1 Theorem 4.1 on the spectral invariance of the symbol class Mv1/2 implies that the ∞,1 −1 ∈ ∞,1 inverse operator T also possesses a symbol in Mv1/2 , i.e., T Op(Mv1/2 ). −1 −1 −1 −1 Now, since T = J1/θT Jθ , we ﬁnd that −1 −1 T = J1/θT Jθ. Applying Theorem 2.4 once again, the symbol of T −1 must be in M ∞,1.Asa consequence, T −1 is bounded on L2(Rd). ∈ ∞,1 2 Rd Since T Op(Mv )andT is invertible on L ( ), it follows that T is also p,q Rd ∈M ≤ ≤∞ invertible on Mμ ( ) for every weight μ v and 1 p, q . 4.2. An isomorphism theorem for localization operators. We now combine all steps and formulate and prove our main result, the isomorphism theorem for time-frequency localization operators with symbols of superfast growth. ∈ 1 Rd ∈M ∈M Theorem 4.3. Let g Mv2w( ), μ w and m v be such that m is radial in each time-frequency coordinate and v satisﬁes (31). Then the localization g p,q Rd p,q Rd ≤ ≤∞ operator Am is an isomorphism from Mμ ( ) onto Mμ/m( ) for 1 p, q . g g Proof. Set T = A1/mAm. We have already established that g g ∞,1 (1) T = A1/mAm possesses a symbol in Mv˜w˜ by Corollary 2.5. 2 Rd (2) T is invertible on Mθ ( ) by Lemma 2.2. p,q Rd These are the assumptions of Theorem 4.2, and therefore T is invertible on Mμ ( ) ∈M ⊆M ≤ ≤∞ g g for every μ w vw and 1 p, q . The factorization T = A1/mAm g p,q Rd p,q Rd g implies that Am is one-to-one from Mμ ( )toMμ/m( )andthatA1/m maps p,q Rd p,q Rd Mμ/m( )ontoMμ ( ).

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g g Now we change the order of the factors and consider the operator T = AmA1/m p,q Rd p,q Rd p,q Rd from Mμ/m( )toMμ/m( ) and factoring through Mμ ( ). Again T possesses ∞,1 2 Rd asymbolinMv˜w˜ and is invertible on M1/θ( ). With Theorem 4.2 we conclude p,q Rd that T is invertible on all modulation spaces Mμ/m( ). The factorization of g g g p,q Rd p,q Rd T = AmA1/m now yields that A1/m is one-to-one from Mμ/m( )toMμ ( ) g p,q Rd p,q Rd and that Am maps Mμ ( )ontoMμ/m( ). g p,q Rd p,q Rd g As a consequence Am is bijective from Mμ ( )ontoMμ/m( )andA1/m is p,q Rd p,q Rd bijective from Mμ/m( )ontoMμ ( ).

5. Consequences for Gabor multipliers and Toeplitz operators on the Bargmann-Fock space 5.1. Gabor multipliers. Gabor multipliers are time-frequency localization operators whose symbols are discrete measures. Their basic properties are the same as for the corresponding localization operators, but the discrete definition makes them more accessible for numerical computations. Let Λ = AZ2d for some A ∈ GL(2d, R) be a lattice in R2d,letg, γ be suitable window functions (as in Proposition 5.1) and m be a weight sequence defined on Λ. Then the Gabor multiplier Gg,γ,Λ is defined to be m g,γ,Λ (33) Gm f = m(λ) f,π(λ)g π(λ)γ. λ∈Λ The boundedness of Gabor multipliers between modulation spaces is formulated and proved exactly as for time-frequency localization operators. See [15] for a detailed exposition of Gabor multipliers and [10] for general boundedness results that include distributional symbols. ∈ 1 Rd Proposition 5.1. Let g, γ Mvw( ),andletm be a continuous moderate weight ∈M g,γ,Λ p,q R2d p,q Rd function m v.ThenGm is bounded from Mμ ( ) to Mμ/m( ). In the following we will assume that the set G(g, Λ) = {π(λ)g : λ ∈ Λ} is a 2 Rd g,g,Λ Gabor frame for L ( ). This means that the frame operator Sg,Λf = M1 = 2 Rd λ∈Λ f,π(λ)g π(λ)g is invertible on L ( ). From the rich theory of Gabor frames ∈ 1 G 2 Rd we quote only the following result: If g Mv and (g, Λ) is a frame for L ( ), 1 ≤ p ≤∞,andm ∈Mv,then 1/p p p p d p | | ∈ R (34) f Mm f,π(λ)g m(λ) for all f Mm( ). λ∈Λ We refer to [17] for a detailed discussion, the proof, and for further references. Our main result on Gabor multipliers is the isomorphism theorem. ∈ 1 Rd G 2 Rd Theorem 5.2. Assume that g Mv2w( ) and that (g, Λ) is a frame for L ( ). ∈M g,g,Λ If m v is radial in each time-frequency coordinate, then Gm is an isomor- p,q R2d p,q Rd ∈M ≤ ≤∞ phism from Mμ ( ) onto Mμ/m( ) for every μ w and 1 p, q . The proof is similar to the proof of Theorem 4.3, and we only sketch the necessary modifications. In the following we fix the lattice Λ and choose γ = g and drop the g,γ,Λ g reference to these additional parameters by writing Gm as Gm in analogy to the g localization operator Am.

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∈ 1 Rd ∈M 1/2 g Proposition 5.3. If g Mv ( ) and m v and θ = m ,thenGm is an 2 Rd 2 Rd isomorphism from Mθ ( ) onto M1/θ( ).

g 2 2 2 Proof. By Proposition 5.1 Gm is bounded from Mθ to Mθ/m = M1/θ, and thus g G f 2 ≤ Cf 2 . m M1/θ Mθ 2 Next we use the characterization of Mθ by Gabor frames and relate it to Gabor multipliers: g | |2 2 (35) Gmf,f = m(λ) f,π(λ)g f 2 . Mθ λ∈Λ g 2 g This identity implies that Gm is one-to-one on Mθ and that Gmf M1/θ is an 2 g 2 equivalent norm on Mθ .SinceGm is self-adjoint, it has dense range in M1/θ, g whence Gm is onto as well. g Using the weight 1/m instead of m and 1/θ instead of θ,weseethatG1/m is an 2 Rd 2 Rd isomorphism from M1/θ( )ontoMθ ( ). Proof of Theorem 5.2. We proceed as in the proof of Theorem 4.3. Deﬁne the g g g g 2 Rd operators T = G1/mGm and T = GmG1/m. By Proposition 5.1 T maps Mθ ( ) 2 Rd to Mθ ( ), and since T is a composition of two isomorphisms, T is an isomorphism 2 Rd 2 Rd on Mθ ( ). Likewise T is an isomorphism on M1/θ( ). ∞,1 R2d As in Corollary 2.5 we verify that the symbol of T is in Mv˜w˜ ( ). Theorem 4.2 then asserts that T is invertible on L2(Rd), and the general spectral invariance p,q Rd implies that T is an isomorphism on Mμ ( ). Likewise T is an isomorphism on p,q Rd Mμ/m( ). This means that each of the factors of T must be an isomorphism on the correct space.

5.2. Toeplitz operators. The Bargmann-Fock space F = F 2(Cd) is the Hilbert space of all entire functions of d variables such that 2 2 −π|z|2 (36) F F = |F (z)| e dz < ∞. Cd Here dz is the Lebesgue measure on Cd = R2d. Related to the Bargmann-Fock space F 2 are spaces of entire functions satisfying weighted integrability conditions. Let m be a moderate weight on Cd and 1 ≤ ≤∞ F p,q Cd p, q . Then the space m ( ) is the Banach space of all entire functions of d complex variables such that q/p q | |p p −pπ|x+iy|2/2 ∞ (37) F F p,q = F (x + iy) m(x + iy) e dx dy < . m Rd Rd Let P be the orthogonal projection from L2(Cd,μ) with Gaussian measure 2 dμ(z)=e−π|z| dz onto F 2(Cd). Then P is given by the formula 2 (38) PF(w)= F (z)eπzwe−π|z| dz. Cd We remark that a function on R2d with a certain growth is entire if and only if it satisﬁes F = PF.

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The classical Toeplitz operator on Bargmann-Fock space with a symbol m is 2 d deﬁned by TmF = P (mF )forF ∈F (C ). Explicitly Tm is given by the formula πzw −π|z|2 (39) TmF (w)= m(z)F (z)e e dz. Cd See [2, 3, 4, 5, 12, 16] for a sample of references. In the following we assume that the symbol of a Toeplitz operator is continuous and radial in each coordinate, i.e., m(z1,...,zd)=m0(|z1|,...,|zd|)forsome Rd R continuous function m0 from + to +.

Theorem 5.4. Let μ ∈Mw,andletm ∈Mv be a continuous moderate weight function such that one of the following conditions is fulﬁlled: (i) Either m is radial in each coordinate (ii) or m is of polynomial type. F p,q Cd F p,q Cd Then the Toeplitz operator Tm is an isomorphism from μ ( ) onto μ/m( ) for 1 ≤ p, q ≤∞. The formulation of Theorem 5.4 looks similar to the main theorem about time- frequency localization operators. In fact, after a suitable translation of concepts, it is a special case of Theorem 4.3. To explain the connection, we recall the Bargmann transform that maps distributions on Rd to entire functions on Cd: 2 2 (40) Bf(z)=F (z)=2d/4e−πz /2 f(t)e−πt e2πt·zdt, z ∈ Cd. Rd If f is a distribution, then we interpret the integral as the action of f on the function 2 e−πt e2πt·z. The connection to time-frequency analysis comes from the fact that the Barg- mann transform is just a short-time Fourier transform in disguise [17, Ch. 3]. As 2 before, we use the normalized Gaussian h(t)=2d/4e−πt as a window. Identifying the pair (x, ξ) ∈ Rd × Rd with the complex vector z = x + iξ ∈ Cd, the short-time Fourier transform of a function f on Rd with respect to h is

πix·ξ −π|z|2/2 (41) Vhf(z)=e Bf(z)e . In particular, the Bargmann transform of the time-frequency shift π(w)h is given by

2 (42) B(π(w)h)(z)=e−πiu·ηeπwze−π|w| /2,w,z∈ Cd,w = u + iη. It is a basic fact that the Bargmann transform is a unitary mapping from L2(R)onto 2 d the Bargmann-Fock space F (C ). Furthermore, the Hermite functions hα,α≥ 0, |α|/2 −1/2 α are mapped to the normalized monomials eα(z)=π (α!) z . Let m(z)=m(z). Then (41) implies that the Bargmann transform B maps p,q Rd F p,q Cd Mm ( ) isometrically to m ( ). By straightforward arguments it follows that p,q Rd the Bargmann transform maps the modulation space Mm ( )ontotheFockspace F p,q Cd m ( ) [17, 28]. The connection between time-frequency localization operators and Toeplitz operators on the Bargmann-Fock space is given by the following statement.

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Proposition 5.5. Let m be a moderate weight function on R2d Cd and set m (z)=m(z). The Bargmann transform intertwines Jm and Tm , i.e., for all ∈ 2 Rd ∈ 1/2 ∗ f L ( ) (or f (S1/2 ) ) we have

(43) B(Jmf)=Tm (Bf). Proof. This fact is well known [8, 12]. For completeness and consistency of notation, we provide the formal calculation. h We take the short-time Fourier transform of Jm = Am with respect to h.On the one hand we obtain that πiu·η −π|w|2/2 (44) Jmf,π(w)h = e B(Jmf)(w) e , and on the other hand, after substituting (41) and (42), we obtain

Jmf,π(w)h = m(z)Vhf(z)Vh(π(w)h)(z) dz R2d 2 (45) = m(z)Bf(z) B(π(w)h)(z)e−π|z| dz Cd 2 2 = eπiu·η m(z) Bf(z)eπzw e−π|z| dz e−π|w| /2. Cd Comparing (44) and (45) we obtain πzw −π|z|2 B(Jmf)(w)= m(z)Bf(z)e e dz = Tm Bf(w). Cd Proof of Theorem 5.4. First assume that (i) holds, i.e., m is radial in each variable. The Bargmann transform is an isomorphism between the modulation space p,q Rd F p,q Cd Mm ( ) and the Bargmann-Fock space m ( ) for arbitrary moderate weight function m and 1 ≤ p, q ≤∞. By Theorem 4.3 the anti-Wick operator Jm is an p,q p,q Rd Rd isomorphism from Mμ ( )ontoMμ/m( ). Since Tm is a composition of three isomorphisms (Proposition 5.5 and (46)), the Toeplitz operator Tm is an isomor- F p,q Cd F p,q Cd phism from μ ( )onto μ/m ( ):

F p,q −→Tm F p,q μ μ/m (46) ↑B ↑B . p,q −→Jm p,q Mμ Mμ/m Finally replace m and μ by m and μ. By using Theorem 3.2 in [21] instead of Theorem 4.3, the same arguments show that the result follows when (ii) is fulﬁlled.

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Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria E-mail address: [email protected] Department of Computer Science, Physics and Mathematics, Linnæus University, Vaxj¨ o,¨ Sweden E-mail address: [email protected]

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