Time-Frequency Analysis and Coorbit Spaces of Operators

We introduce an operator valued Short-Time Fourier Transform for certain classes of operators with operator windows, and show that the transform acts in an analogous way to the Short-Time Fourier Transform for functions, in particular giving rise to a family of vector-valued reproducing kernel Banach spaces, the so called coorbit spaces, as spaces of operators. As a result of this structure the operators generating equivalent norms on the function modulation spaces are fully classified. We show that these operator spaces have the same atomic decomposition properties as the function spaces, and use this to give a characterisation of the spaces using localisation operators.


Introduction
In time-frequency analysis, the modulation spaces M p,q m pR d q, first introduced by Feichtinger in 1983 [16], play a central role, where they define spaces of functions with certain desirable time-frequency decay.In particular the Feichtinger algebra, M 1 pR d q or S 0 pR d q [14] [13], gives well concentrated functions in the time-frequency sense, which are for many purposes the ideal atoms for Gabor analysis.The modulation spaces are usually defined in terms of the Short-Time Fourier Transform (STFT), namely as the spaces M p,q m pR d q :" tψ P S 1 pR d q : ´żR d ´żR d |V φ 0 ψpzq| p mpx, ωq p dx ¯q{p dω ¯1{q ă 8u, where φ 0 is the Gaussian.Modulation spaces and their various generalisations have been studied extensively, and surveys and monographs can be found in [25] [6] [22].The properties and utility of these function spaces are too broad to hope to cover, but of particular interest to our work is that these spaces are the coorbit spaces [17] [18] of the projective unitary representation of the reduced Weyl-Heisenberg group, and as such have (among others) the following properties: (1) All g P L 2 pR d q that satisfy the condition V g g P L 1 v pR 2d q, generate the same modulation spaces M p,q m pR d q as windows, and their norms are equivalent.
(2) (Correspondence Principle) Given an atom g as above, there is an isometric isomorphism M p,q m pR d q -tF P L p,q m pR 2d q : F " F 6V g gu (where 6 is the twisted convolution discussed below), given by V g .Note that the later are reproducing kernel Banach spaces.There is a vast body of contributions to the theory of coorbit spaces, e.g.[4,8,24].In this work we examine spaces of operators exhibiting similar properties, by introducing an STFT with operator window and argument, returning an operator-valued function on phase space.One motivation comes from [12], where local structures of a data set D " tf 1 , ..., f N u were identified via mapping the data points of functions f i on R d to rank-one operators f i b f i , and constructing the data operator S D " ř N i"1 f i b f i .Hence, it would be of interest to compare to data sets D and D 1 via its respective data operators S D and S D 1 .Another source of inspiration is the work [26], where operator analogues of the Schwartz class of functions and of the space of tempered distributions have been introduced and their basic theory has been developed along the lines of the function/distribution case.
The concept of an STFT for operators is not a new one.In [5], the authors consider the wavelet transform for the representation πpwqbπpzq on HS " L 2 pR d q b L 2 pR d q to examine kernel theorems for coorbit spaces.This entails using the standard scalar-valued construction for the coorbit spaces defined by the wavelets transform, giving different spaces to our approach.On the other hand nor are vector-valued reproducing kernel Hilbert spaces in time-frequency analysis a new concept.In [2] and [1] an STFT is constructed for vectors of functions, which results in a direct sum of Gabor spaces.Our work differs from these in that windows, arguments and resulting output of the operator STFT are all operators.
In [34], the author introduced an equivalent notion of a STFT with an operator window, given by V S ψpzq :" Sπpzq ˚ψ (1) for some appropriate operator S and function ψ.In particular, the question was considered of which operators would define equivalent norms on M p,q m pR d q under this STFT, that is, for which operators }ψ} M p,q m -}Sπpzq ˚ψ} L p,q m pR 2d ;L 2 q .In further work by Guo and Zhao [23], some equivalent conditions for equivalence were given.In both works a class of operators with adjoints in a certain class of nuclear operators was discussed, along with the open question in the latter of whether these operators exhausted all possible operators generating equivalent norms on M p,q m pR d q.In this work we present an extension of the operator window STFT (1), which acts on operators instead of functions.We initially define such a transform for S, T P HS in the following manner: We examine the behaviour of this transform, e.g.Moyal's identity, paying particular attention to the spaces it produces as images.In this respect the first result of this paper demonstrates a parallel to the STFT of functions, regarding the reproducing structure of the image of the Hilbert space of Hilbert-Schmidt operators: Theorem 1.2.For any Hilbert-Schmidt operator S, the space defined by V S pHSq :" tV S T pzq : T P HSu is a vector-valued uniform reproducing kernel Hilbert space as a subspace of the Bochner-Lebesgue space L 2 pR 2d ; HSq.
Motivated by this, we extend the reproducing properties of this space to the "coorbit spaces", and consider the spaces A v :" tS P HS : V S S P L 1 v pR 2d ; HSqu, and M p,q m :" tT P S 1 : V S T P L p,q m pR 2d ; HSqu, where S 1 are operators with Weyl symbols in S 1 and S P A v , to derive the result Theorem 1.3.For any S P M 1 v , we have an isometric isomorphism M p,q m -tΨ P L p,q m pR 2d ; HSq : Ψ " Ψ6V S Su under the mapping the twisted convolution 6 is to be defined in Section 2.3.Furthermore, for all S P A v the resulting spaces coincide, and the associated norms are equivalent.The dual space of M p,q m is M p 1 ,q 1 1{m , where 1 p `1 p 1 " 1, 1 q `1 q 1 " 1 with the usual adjustment for p, q " 1, 8. As a corollary of the coorbit structure and independence of windows, we characterise operators satisfying the equivalent norm condition; Corollary 1.4.The operators which define equivalent norms on the spaces M p,q m pR d q by }S ˚πpzq ˚ψ} L p,q m pR 2d ;L 2 pR d qq or every 1 ď p, q ď 8 and v-multiplicative m, are precisely the admissible operators A v :" tS : V S S P L 1 v pR 2d ; HSqu, We finally consider the atomic decomposition of operators in the M p,q m , which follows from the same arguments as the function case given the coorbit structure.Using this we can characterise the spaces using localisation operators: Corollary 1.5.Let φ P L 2 pR d q be non-zero and h P L 1 v pR 2d q be some nonnegative symbol satisfying 1{v belongs to M p,q m if and only if ␣ A φ h πpλq ˚T ( λPΛ P l p,q m pΛ; HSq.
where Λ " αZ ˆβZ is some full rank lattice.

Preliminaries
2.1.Time-Frequency Analysis Basics.While coorbit spaces are defined in general for integrable representations of locally compact groups, modulation spaces of functions and the spaces discussed in this work arise from the particular case of the time-frequency shifts πpzq, the projective unitary representation of the reduced Weyl-Heisenberg group on the Hilbert space L 2 pR d q.Such shifts can be defined as the composition of the translation operator T x : f ptq Þ Ñ f pt ´xq, and the modulation operator M ω : f ptq Þ Ñ e 2πiωt f ptq, by the identity where z " px, ωq P R 2d .Direct calculations show that πpzq is unitary on L 2 pR d q, and that we have πpzqπpz 1 q " e ´2πiω 1 x πpz `z1 q πpzq ˚" e ´2πixω πp´zq.
The Short-Time Fourier Transform (STFT) for functions is then defined, for two functions f, g P L 2 pR d q, by The window function g is usually chosen to have compact support, or be concentrated around the origin, such as in the case of the normalised Gaussian φ 0 ptq " 2 d{4 e πt 2 .For f, g P L 2 pR 2d q, V g f is uniformly continuous as a function in L 2 pR 2d q, which will be instructive when considering reproducing kernel Hilbert spaces later.One has for the STFT Moyal's Identity (see for example Theorem 3.2.1 of [22]), giving an understanding of the basic properties of the STFT in terms of its window: Lemma 2.1.(Moyal's Identity) Given functions f 1 , f 2 , g 1 , g 2 P L 2 pR d q, we have V g 1 f 1 , V g 2 f 2 P L 2 pR 2d q, and in addition: As a direct consequence, we have that for any g P L 2 pR d q such that }g} L 2 " 1, the map V g : L 2 pR d q Ñ L 2 pR 2d q is an isometry.As such, we can consider the inverse mapping.Rearranging Moyal's identity shows the reconstruction formula for any g P L 2 pR d q with }g} " 1.A direct calculation then shows that the adjoint V g is given by where the integral can be interpreted in the weak sense, and so from the reconstruction formula 2.2.Weight functions and mixed-norm spaces.We begin by defining a sub-multiplicative weight v as a non-negative, locally integrable function on phase space R 2d satisfying the condition vpz 1 `z2 q ď vpz 1 qvpz 2 q for all z 1 , z 2 P R 2d .As a direct result, vp0q ě 1.A v-moderate weight m is then a non-negative, locally integrable function on phase space such that mpz 1 `z2 q ď vpz 1 qmpz 2 q for all z 1 , z 2 P R 2d .As a particular consequence, we have for such a v, m In this work we consider weights of at most polynomial growth.We define the weighted, mixed-norm space L p,q m pR 2d q, for 1 ď p, q ă 8, as the functions for which the norm }F } L p,q m :" ´żR d ´żR d |F px, ωq| p mpx, ωq p dx ¯q{p dω ¯1{q is finite.In the case where p or q is infinite, we replace the corresponding integral with essential supremum.For such spaces we have the duality pL p,q m pR 2d qq 1 " L p 1 ,q 1 1{m pR 2d q, where 1 p `1 p 1 " 1, 1 q `1 q 1 " 1.Further details on weights and mixed-norm spaces can be found in chapter 11, [22].In this work we consider discretisation over the full rank lattice Λ " αZ d ˆβZ d .An arbitrary lattice Λ " AZ 2d , A P GLp2d; Rq can also be used, but the notion of mixed-norm becomes less clear.We define the mixed-norm weighted sequence space l p,q m pΛ; HSq as the sequences a pk,lq such that }a} l p,q m pΛ;HSq :" ´ÿ nPZ d `ÿ kPZ d mpαk, βlq p }a αk,βl } p HS ˘q{p ¯1{q ă 8.
The Wiener Amalgam spaces introduced in [15] provide the required framework for sampling estimates on the lattice.To that end we define for a given function Ψ : R 2d Ñ HS the sequence a Ψ pk,lq " ´ess sup x,ωPr0,1s d }Ψpx `k, ω `lq} HS ¯pk,lq .
While stated for scalar-valued functions in [22], the same argument gives the vector-valued case.

2.3.1.
Vector-Valued RKHS.We recall definitions and identities in this section in terms of vector-valued reproducing kernel Hilbert spaces, following the formalism of Paulsen and Raghupathi in chapter 6 of [30].The familiar scalar case follows simply by considering the vector space which functions take their values to be C. Definition 2.4.Let C be a Hilbert space, and X some set.We denote by FpX, Cq the vector space of C-valued functions under the usual pointwise sum and scalar multiplication.A subspace H Ď F is a C-valued reproducing Kernel Hilbert Space (RKHS) if it is a Hilbert space, and for every x P X, the evaluation map E x : f Ñ f pxq is a bounded operator.If the set tE x u xPX is uniformly bounded in norm, then H is referred to as uniform.
Since H is a Hilbert space, it follows from Riesz' representation theorem that for each E x , there is some so unlike in the general Hilbert space setting, we have pointwise bounds in terms of norms in the RKHS setting.The kernel function K : X ˆX Ñ LpCq is defined as Kpx, yq " E x E ẙ , and has the property Kpx, yq " Kpx, yq ˚.The kernel function uniquely defines the RKHS, that is to say given two RKHS' H 1 , H 2 , if K 1 px, yq " K 2 px, yq then H 1 " H 2 and } ¨}H 1 " } ¨}H 2 , and vice versa.
This result can be deduced by noting that V g is an isometry onto its image, then proceeding with the adjoint as defined above.
2.3.2.Twisted Convolutions.Reproducing properties of Gabor spaces are intimately connected to the twisted convolution, which is defined in terms of the 2-cocycle of πpzq, which we define as cpz, z 1 q " e ´2πix 1 pω´ω 1 q , such that πpzq ˚πpz 1 q " cpz, z 1 qπpz `z1 q ˚.We define the twisted convolution for a Lebesgue-Bochner space (for details see for example [9]).The twisted convolution presented here swaps the arguments, but this is only in order to fit our construction of the operator STFT with the standard notation in coorbit theory.Definition 2.6.Given two operator-valued functions F, H P L 2 pR 2d ; HSq, we define the twisted convolution 6 as Hpx ´yqF pyqcpx ´y, yq dy, where the integral can be interpreted in the sense of a Bochner integral.
In the concrete setting of Gabor spaces, a direct calculation shows for functions f 1 , f 2 , g 1 , g 2 P L 2 pR 2d q, that V g 1 f 1 6V g 2 f 2 " xf 2 , g 1 yV g 2 f 1 .Clearly then for some F P V g pL 2 q, the identity F 6V g g " F holds when }g} L 2 " 1, however a fundamental result of coorbit theory is that the converse also holds, giving the following: Proposition 2.7.Given some g P L 2 pR d q with }g} L 2 " 1, a function F P L 2 pR 2d q is in V g pL 2 q if and only if F 6V g g " F .
We note an application of weighted, mixed-norm Young's inequality to Lebesgue-Bochner spaces of Banach algebras to be used in the sequel.A proof of the scalar valued case can be found for example in Proposition 11.1.3 of [22], the vector valued case follows by the same argument.
Lemma 2.8.Given functions F P L 1 v pR 2d ; HSq and H P L p,q m pR 2d ; HSq we have m , where v is some sub-multiplicative function and m a v-moderate weight, and C m,v a constant depending on v and m.
2.4.Modulation Spaces.We begin by considering the space M 1 v pR d q.For a sub-multiplicative v, we define the modulation space as functions whose image under the STFT with Gaussian window is in qu.Such a space is always non-empty, as φ 0 itself is contained in it, and for weights of polynomial growth it contains the Schwartz functions S .In addition, it is closed under pointwise multiplication, time-frequency shifts, and is a Banach space under the norm }f } The unweighted M 1 pR d q is Feichtinger's algebra, which has been studied extensively and provides for many avenues of time-frequency analysis the ideal set of test functions.We refer to the early paper [14] and recent survey [25] for more details on the space.General modulation spaces are then defined, for any v-moderate weight m, by M p,q m pR d q :" tf P pM 1 v pR d qq 1 : V φ 0 f P L p,q m pR 2d qu, with the associated norm }f } M p,q m " }f } L p,q m For any g P M 1 v pR d q, the space tf P pM 1 v pR d qq 1 : V g f P L p,q m pR 2d qu is equal to the space M p,q m pR d q and the associated norms are equivalent.It is not hard to see that M 2,2 pR d q " L 2 pR d q, from the properties of the STFT with window φ 0 P L 2 pR 2d q.The modulation spaces form coorbit spaces of the unitary representation π, and as such have the property: Theorem 2.9.(Correspondence Principle) Given some g P M 1 v pR d q, for every 1 ď p ď 8, the STFT defines an isomorphism V g : M p,q v pR d q Ñ tF P L p,q v pR 2d q : F 6V g g " F u. Remark 2.10.In this work we consider the case of the operator STFT V S .One might therefore ask why we do not refer simply to an operator modulation space.We believe this would be misleading, since the term modulation space refers to the construction of the spaces by the M 1 pR d q condition ş Ĝ }M ω f ˚f } 1 dω ă 8.We do not work with the analogous concept of modulation for operators, so we choose to refer to them as coorbit spaces.
Although not coorbit spaces in the "strict sense" [36], since the elements are not in the representation space of π, they can be considered as generalised coorbit spaces in the sense that instead of our transforms being a functional, they are now simply maps between spaces of operators, for example HS Ñ HS.
2.5.Spaces of Operators.for any orthonormal basis te n u nPN of L 2 pR d q.For operators satisfying this condition, the sum ř nPN xT e n , e n y in fact converges for all orthonormal bases to the same value, the trace of T , given by trpT q.The set of such operators is then a Banach space when equipped with the norm }T } S 1 " trp|T |q.The Hilbert-Schmidt operators HS, are the operators HS :" tT P LpL 2 pR d qq : The space HS is a Hilbert-Schmidt space with the inner product xS, T y HS " trpST ˚q, and contains S 1 as a proper ideal.We will often use that every compact operator, and therefore every Hilbert-Schmidt and trace class operator, admits a spectral decomposition where λ n are the singular values of S, tψ n u nPN and tϕ n u nPN are orthonormal sets and the sum converges in operator norm.Both these spaces are Banach algebras with their respective norms and two-sided ideals in LpL 2 pR d qq, with S 1 Ă HS Ă LpL 2 pR d qq.The further Schatten class operators, S p , are defined by the decay of their singular values; S p :" tT P LpL 2 pR d qq : tλ n u nPN P l p u where λ n are again the singular values of T .Clearly HS " S 2 .We also introduce a space of nuclear operators, a concept which generalises the concept of trace to operators between Banach spaces.In particular for two Banach spaces X, Y , the nuclear operators N pX, Y q are the linear operators T which have an expansion T " ř n y n bx n , where These operators become a Banach space when endowed with the norm }T } N pX,Y q " inf ř n }y n } Y }x n } X 1 where the infimum is taken over all possible decompositions of T .In our case we are interested in the nuclear operators N pL 2 pR d q; M 1 v pR d qq.Such operators may be defined as the projective tensor product N pL 2 pR d q; M 1 v pR d qq :" M 1 v pR d q bπ L 2 pR d q, the completion of the algebraic tensor product M 1 v pR d qbL 2 pR d q with respect to the nuclear norm Finally we introduce the Schwartz operators S, as the space of bounded integral operators with kernel k P S pR 2d q.Such operators form a Frechet space as detailed in [26], and the topological dual S 1 consists of integral operators with kernels in S 1 pR 2d q, which by the Schwartz kernel theorem is the space of operators from S pR 2d q to S 1 pR 2d q.For polynomial sub-multiplicative weight v, we use the sequence of inclusions S Ă N pL 2 pR d q; M 1 v pR d qq Ă HS Ă S 1 .2.5.2.G-frames for Operators.In the operator setting, we will consider gframes as introduced in [35] as an analogue to frames in the function setting.In particular, given a Hilbert space U, and a sequence of Hilbert spaces tV i u iPI , then a sequence of operators tS i P LpU; V i qu iPI is called a g-frame of U with respect to tV i u iPI if there exists positive constants A, B such that the g-frame condition holds for all u P U. We call tS i u iPI a tight frame when A " B, and a Parseval frame when A " B " 1.In our work we consider the case where V i coincide for all i.When the g-frame condition holds, the g-frame operator is positive, bounded and invertible on U.In [33], g-frame operators of the type ÿ λPΛ πpλqS ˚Sπpλq for some lattice Λ, were considered on the Hilbert space L 2 pR d q.In this work, we say an operator S P LpL 2 pR d qq generates a Gabor g-frame if tS ˚πpλq ˚uλPΛ is a frame for HS.Proposition 2.11.If S P LpL 2 pR d qq generates a Gabor g-frame of HS for some lattice Λ, then S P HS.
This follows from the same argument as Proposition 5.7 in [33], taking a rank-one T .
For S P HS which generates a Gabor g-frame, we define the analysis operator C S : HS Ñ l 2 pΛ; HSq by C S T " tS ˚πpλq ˚T u λPΛ , and the synthesis operator D S : l 2 pΛ; HSq Ñ HS by If S generates a Gabor g-frame, then general g-frame theory [35] tells us there exists a canonical dual frame t Sλ u λPΛ :" tS ˚πpλq ˚O´1 S u λPΛ since T " O ´1 S O S T " O S O ´1 S T .S can be shown to be a Gabor g-frame generated by pS ˚O´1 S q.We say in general that two operators S, T P LpHSq generate dual Gabor g-frames if S and T generate Gabor g-frames, and O S,T :" D S C T " I HS .
2.6.Quantum Harmonic Analysis.As a final prerequisite we present some theorems of quantum harmonic analysis, based on the convolutions introduced by Werner in [37], and recently applied to time-frequency anlysis in [27] [28] [29], where it is used to generalise known results and provide more concise proofs by extending the mechanics of harmonic analysis to operators.We will on occasion use the framework of quantum harmonic analysis to simplify a proof or give an alternative framing.Convolutions between operators and functions are defined in the following manner; Definition 2.12.For f P L p pR 2d q, S P S q and T P S p , where 1 p `1 q " 1`1 r , convolutions are defined by where α z pSq " πpzqSπpzq ˚is a representation of the Weyl-Heisenberg group on HS and Ť " P T P where P is the parity operator.The first integral is to be interpreted as a Bochner integral.
We will use a generalised version of Moyal's identity from [37]; Lemma 2.13.(Generalised Moyal's Identity) For two operators S, T P S 1 , the mapping z Þ Ñ Sα z pT q is integrable over R 2d , and ż R 2d Sα z pT q dz " trpSqtrpT q.
Taking rank one operators returns precisely the original Moyal's identity, hence the name.We also note that the above holds when T is replaced with Ť , since trpT q "trpP T P q.We also make frequent use of the fact that for S P S 1 ; which can be seen by using the spectral decomposition of S and the reconstruction formula for V g .

An Operator STFT
We start by defining the operator valued STFT.Definition 3.1.(Operator STFT) For two HS operators S, T on L 2 pR d q, the operator short-time Fourier transform, V S T , is given by (8) V S T pzq " S ˚πpzq ˚T .
The operator STFT thus defines an operator valued function in phase space.We will see that this operator valued function is in many respects an analogue to the scalar function of the function STFT.To motivate such a definition, we consider the following: Example 3.2.For operators S " ř n g n b e n and T " ř n f n b e n with f n , g n P L 2 pR d q and te n u n some orthonormal basis in L 2 pR d q; Remark 3.3.Here and in the sequel, we will often consider an operator S " ř n f n b e n where only the e n are assumed to be orthonormal.This is done because we will later consider different norms on the f n .In the above example we could of course assume the f n to have }f n } L 2 " s n , where s n are the singular values of S.
This definition is clearly equivalent to the definition in [34], [23] in the case of a rank one T " ψ b ξ, where we have V S T " pS ˚πpzq ˚ψq b ξ, except that we consider the adjoint S ˚.This adjustment is to make formulae in the sequel cleaner, and we note that there is no material difference in the two formulations.The STFT can thus be considered to encode information about time frequency correlations over functions.
Example 3.4.Let A φ 1 ,φ 2 f and A ψ 1 ,ψ 2 g be standard single window localisation operators given by where f, g P L 2 pR 2d q and φ i , ψ i P L 2 pR d q.The operator STFT of A ψ 1 ,ψ 2 g with window A φ 1 ,φ 2 f is then This expression has an intuitive interpretation; if windows ψ 1 and φ 1 are concentrated in time-frequency around the origin, then the inner product in the integrand is negligible outside of the region around z " z 2 ´z1 .To illustrate this we consider the simple situation where f, g are characteristic functions, and all windows are the Gaussian φ 0 : where A Ω i :" A φ 0 ,φ 0 χ Ω i .Fig. 1 shows some simple domains Ω i in the timefrequency plane.In Fig. 2, the resulting Hilbert-Schmidt norm of the operator STFTs V A Ω i A Ω j are shown as a function of z: These examples illustrates how the Hilbert-Schmidt norm of the operator STFT acts, but we also have the interpretation of πpz 1 qφ 0 b πpz 2 qφ 0 , as the operator sending time-frequency energy of a function from around z 2 to around z 2 ´z.Upon taking the taking the Hilbert-Schmidt norm, we recover the total correlation function from [12]; Hence the structure of the resulting operator can be seen to provide more information regarding the correlations within the dataset, as it relates where in the dataset the correlation occurs, for example on the diagonal versus off.To see this we compare two operators with identical total correlation functions, but one generated by functions determined by stationary process, while the other is generated by functions drawn from a non-stationary process.Both operators are of the form where e i form an orthonormal basis, and f i are of the form where a i is a constant from a random normal distribution, g i a Bartlett-Hann window translated by a random x from a normal distribution.The length of the signal is 200, and so the resulting matrix S i has dimensions 200ˆ200.For   the operator S 1 , the frequencies freq i are given by a base frequency with the addition of random noise from both a normal and sinoidal distribution.The operator S 2 on the other hand, is generated by the same base frequency, with random noise from a normal distribution, but with an i-dependent modulation.The two operators have an identical total correlation function, shown in Fig. 3, but comparing the operator-value of the STFT for different z's shows the non-stationary structure of the function data set generating the operator S 2 , Fig. 5, when compared to S 1 Fig. 4. In this respect the operator STFT can be seen to reflect the structure of an ordered data set, such as a functional time series.
We collect some simple properties of the operator STFT: Proposition 3.6.For operators Q, R, S, T P HS; (1) V S T pzq " e ´2πiωx pV T Sp´zqq

HS
Proof.The first claim is merely a restatement of the property πpzq ˚" e ´2πixω πp´zq, and the third a special case of the second.To prove the second claim; pT R ˚q ‹ pP QS ˚P q dz " xQ, Sy HS xT, Ry HS where we have used Lemma 2.13 in moving from the third to fourth line.□ In particular, the third statement gives us that V S : HS Ñ L 2 pR 2d ; HSq, and the mapping is continuous and injective.It is then natural to consider the Hilbert space adjoint, V S : L 2 pR 2d ; HSq Ñ HS, which is given by πpzqSΨpzq dz for Ψpzq P L 2 pR 2d ; HSq.The integral can be interpreted in the weak sense in HS.This can be seen directly; xT, πpzqSΨpzqy HS dz.
The operator STFT and its adjoint shares the reconstruction property with the function case, namely where we use (7).We have as a result V S V S " I HS (9) On the diagonals we have precisely the reproducing kernels of the scalarvalued Gabor spaces with windows g n , that is to say kernels of the projections V gn V gn , but we have in addition the off-diagonal terms corresponding to the kernels of the maps V gn V gm .As a general property of RKHS', we have the inclusion V S pHSq Ă L 2 pR 2d ; HSq X L 8 pR 2d ; HSq, since }V S pT qpzq} 2 HS ď xV S pT q, E z E z V S pT qy L 2 pR 2d ;HSq " }V S pT q} 2 L 2 pR 2d ;HSq .

4.1.
Characterisation from Twisted Convolution.In an analogue way to the characterisation of Gabor space in terms of the twisted convolution, we can characterise the RKHS V S pHSq by the equivalent condition.
Proof.On the one hand we have that for Q, R, S, T P HS; S ˚πpz ´z1 q ˚RQ ˚πpz 1 q ˚T e ´2πixpω´ω 1 q dz 1 " S ˚πpzq ˚żR 2d πpz 1 qRQ ˚πpz 1 q ˚dz 1 T " xR, Qy HS V S T pzq, (12) where the last inequality follows from (7), and hence the one direction follows in the case Q " R " S. On the other, from (11), Kpz, z 1 qΨpz 1 q dz 1 " `PS Ψ ˘pzq " Ψpzq implies Ψ P V S pHSq.□ 4.2.Toeplitz operators.With a RKHS structure, it is natural to consider what the corresponding Toeplitz operators on the space look like.Toeplitz operators are of the form T f " P V M f , that is to say a pointwise multiplication by some f P L 8 pR 2d q, followed by a projection back onto the RKHS.
In the case of Gabor spaces these are precisely the localisation or anti-Wick operators, which are accordingly also called Gabor-Toeplitz operators [20].
Considering the Toeplitz operators on V S pHSq, we have operators of the type where f P L 8 pR 2d q and f ¨VS T is pointwise multiplication.We then define the unitarily equivalent operator ΘpT f q :" V S T f V S on HS: and hence Toeplitz operators in the operator case correspond to the composition with the mixed-state localisation operators discussed in [28].

Coorbit Spaces for Operators
From the previous section, we have a characterisation of the space V S pHSq.We now turn to other classes which can be similarly characterised.In particular, from Proposition 3.6 the Hilbert-Schmidt operators are precisely the operators tT P LpL 2 pR d qq : V S T P L 2 pR 2d ; HSqu for S P HS, similarly to the function case of L 2 pR d q " M 2 pR d q.We therefore set out to define what we refer to as operator coorbit spaces.In the sequel, vpzq will be a submultiplicative weight function of polynomial growth on phase space, and mpzq will be a v-moderate weight function on phase space.

The M 1
v case.In a similar vein to the function case we define the admissible operators, for a weight function v, to be A v :" tS P HS : V S S P L 1 v pR 2d ; HSqu.(13) Example 5.1.Clearly any rank one operator which can be written as T " f b ψ, where f P M 1 v pR d q and ψ P L 2 pR d q, is in A v .
We set S 0 " φ 0 b φ 0 , and define the space ;HSq , and we denote the unweighted version vpzq " 1 by M 1 .Remark 5.2.Considering }pφ 0 b φ 0 qπpzq ˚T } HS , it is easy to see how the M 1 v condition (and later the M p,q m conditions) can be seen to measure the time-frequency localisation of an operator.In this case, the M 1 v condition is simply a measure of how time-frequency translations of φ 0 decay as arguments of T ˚: ş vpzq}T ˚pπpzqφ 0 q} L 2 dz.Following this line of reasoning, we consider appropriate localisation operators of the type in Example 3.4: Example 5.3.For a localisation operator A ψ h , if h P L 1 pR 2d q and ψ P M 1 pR d q, then A ψ h P M 1 ; ż |V ψ φ 0 pz ´z1 q| dz dz 1 .
where we have used that }V S T pzq} HS ď }S} HS }T } HS for every z.We can hence decompose every T P M 1 v as T " ř ně0 f n b e n for some orthonormal system te n u n and orthogonal tf n u n .The M 1 v condition Equation ( 14) is then equivalent to vpzq}V φ 0 f n pzq} l 2 pNq dz ă 8u.
Noting that for each n, we find that we assume without loss of generality that }g} L 2 " 1 for all n.Here neither the f n or g n are necessarily orthogonal.We have that

It then follows that
since we assumed }g n } L 2 " 1.The nuclear condition thus gives that ż We conclude that N pL 2 pR d q; M 1 v pR d qq Ă M 1 v □ Remark 5.5.Any T P M 1 v can be written in the form , where }e n } L 2 " 1 for all n, and t}f n } M 1 v u n P l 2 .This follows from the inequality As a result, for the unweighted case, S P M 1 ùñ SS ˚P N pM 1 pR 2 q; M 1 pR 2 qq, or alternatively σ SS ˚P M 1 pR 2d q where σ SS ˚is the Weyl symbol of SS ˚ [19].
Remark 5.6.An operator T " ř n f n b e n in the space M 1 v also satisfies the condition V S 0 T P L 1 pR 2d ; S p q, since V S 0 T takes the values of rank one operators, and so all Schatten class norms coincide.However, as we will later see, using the Hilbert-Schmidt norm is required when considering results for a wider class of window operators.Hence for the sake of consistency we define the operator coorbit spaces in terms of the Hilbert-Schmidt norm.
As a corollary of Claim 5.4, operators T P M 1 v pR d q bπ M 1 v pR d q, and in the case of polynomial growth of v the Schwartz operators S, are contained in M 1 v .We will use this to give a suitably large reservoir for defining general coorbit spaces.5.2.The general M p,q m case.We then define the operator coorbit spaces for 1 ď p, q ď 8 and v-moderate weight m by M p,q m :" tT P S 1 : S 0 πpzq ˚T P L p,q m pR 2d ; HSqu.with norms }T } M p,q m " }V S 0 T } L p,q m .Example 5.7.As in the M 1 v case, any rank one operator which can be written as T " f b ψ, where f P M p,q v pR d q and ψ P L 2 , is in M p,q v .
Remark 5.8.Since we restrict our focus to weights of polynomial growth, the Schwartz operator dual is a sufficiently large reservoir, although if we wished to extend to a larger class of weights this may fail.For weights of exponential growth one has to use ultradistributions [31,7] and it seems to be a promising topic for future research to study these kind of objects in our setting, too.
We use the notation V S T pzq " S ˚πpzq ˚T for S P M 1 v and T P M p,q m , and similarly V S Ψ " ş R 2d πpzqSΨpzqdz for Ψ P L p,q m pR 2d ; HSq.The map V S is injective, as for any non-zero R P M p,q m there exists some f P L 2 pR d q such that Rf is non-zero, and so the injectivity of V S follows from the properties of the function STFT.The M p,q m spaces are clearly closed under addition and scalar multiplication.To show that they are in fact Banach spaces, we use the following lemma: Lemma 5.9.For 1 ď p ď 8 and S P A v , the map V S V S is a bounded operator on L p,q m pR 2d ; HSq, and if }S} HS " 1 then its restriction to V S pM p,q m q is the identity.
Proof.We begin by noting that for Ψ P L p,q m pR 2d ; HSq; Hence from Lemma 2.8; and so V S V S is bounded, since S P A v .For V S T P L p,q m pR 2d ; HSq, as in the HS case we observe α z pS ˚Sq dz T " V S T. □ Corollary 5.10.For 1 ď p ď 8 and S P A v , V S is a bounded map from L p,q m pR 2d ; HSq to M p,q m .Proposition 5.11.For 1 ď p ď 8, 1 ď q ď 8, and v-moderate m, M p,q m is a Banach space.
Proof.We consider a Cauchy sequence tT n u nPN Ă M p,q m .The sequence tV S 0 T n u nPN is a Cauchy sequence in L p,q m pR 2d q by definition of the norm, and since L p,q m pR 2d ; HSq is a Banach space we denote the limit of this sequence Ψ.From Corollary 5.10 V S0 Ψ P M p,q m and T n Ñ V S0 Ψ by boundedness, so M p,q m are Banach spaces.□ As in the function case we have the embedding of our spaces: Claim 5.12.For 1 ď p ď p 1 ď 8, 1 ď q ď q 1 ď 8, and mpzq ě m 1 pzq , M p,q m Ă M p 1 ,q 1 m 1 .This follows from the reproducing formula for M p,q m and the previous lemma; V S 0 V S0 V S 0 T pzq " S 0 πpzq ˚VS 0 V S 0 T .Hence V S 0 M p,q m Ă L p,q m 1 pR 2d ; HSqX L 8 m 1 pR 2d ; HSq and the claim follows.5.3.Equivalent Norms.The twisted convolution structure can be used to show that as in the function case, different operators in M 1 v generate the same M p,q m spaces, with equivalent norms.Proposition 5.13.Given some R P M 1 v , the space tT P HS : V R T P L p,q m pR 2d ; HSqu is equal to the space M p,q m , and the associated norms are equivalent.
Proof.Given R P M 1 v , and T P M p,q m , we aim to show that V R T P L p,q m pR 2d ; HSq.To that end we have from Lemma 2.8 that where C v,m is the v-moderate constant of m, and we have used Proposition 3.6(i).We have also used the formula V Q T 6V S Rpzq " xR, Qy HS V S T pzq, which we initially defined only for T P HS.However examining the argument confirms we are justified in using this for general T .Hence M p,q m Ă tT P HS : V R T P L p,q m pR 2d ; HSqu.Conversely, repeating the above argument with T such that V R T P L p,q m pR d q gives the reverse inclusion.Equivalence of norms is also clear from these symmetric arguments, namely v follows from a similar argument as above.Given some T P A v such that xS, T y HS ‰ 0; In the case that xS, T y HS " 0, we simply take some R P M 1 v with xS, Ry HS ‰ 0 and xT, Ry HS ‰ 0 and repeat the above expansion twice with respect to R to derive □ Corollary 5.15.A Hilbert-Schmidt operator S belongs to the space A v if and only if the following norm equivalence holds: For every 1 ď p, q ď 8, v-multiplicative m.
Proof.The M p,q m pR d q condition }V φ 0 f } L p,q m pR 2d q ă 8 is equivalent to the M p,q m condition }V S 0 pf b φ 0 q} L p,q m pR 2d ;HSq ă 8. From Proposition 5.13, all S P A v determine equivalent norms on these spaces.Conversely for any operator S satisfying Equation (15), for all 1 ď p, q ď 8 and all v-multiplicative m, satisfies }V S pf bφ 0 q} L p,q m pR 2d ;HSq ă 8, and in particular }V S S 0 } L 1 v pR 2d ;HSq ă 8, so S must be in A v by Corollary 5.14.□

Duality.
To show the duality property pM p,q m q 1 -M p 1 ,q 1 1{m where 1 p `1 p 1 " 1 and similarly for q, we follow a similar approach to the function case proof in [22].We will however need a result of [21] for Lebesgue-Bochner spaces: Lemma 5.16.For a Banach space B, and σ-finite measure space pΩ, A, µq, B has the Radon-Nikodym property (RNP) if and only if L p pΩ; Bq 1 " L q pΩ; B 1 q with dual action xa, a ˚yB,B 1 " ż Ω a ˚paq dµ where 1  p `1 q " 1 for 1 ď p ă 8.
Proposition 5.17.For S P A v and 1 ď p ă 8, we have the duality identity pM p,q m q 1 -M p 1 ,q 1 1{m with the dual action given by xT, xV S T pzq, V S Rpzqy HS dz.
Proof.On the one hand, the inclusion M p 1 ,q 1 1{m Ă pM p,q m q 1 is clear from Hölder's inequality for weighted mixed norm spaces; ˇˇż R 2d xV S T pzq, V S Rpzqy HS dz ˇˇď }T } M p,q m }R} M p 1 ,q 1 1{m .
To demonstrate the converse, take R P pM p,q m q 1 .The composition R :" R ˝VS then defines a functional on L p,q m pR 2d ; HSq, by Corollary 5.10.There exists then some Θpzq P L p 1 ,q 1 1{m pR 2d ; HSq, due to Lemma 5.16, such that xΨpzq, Θpzqy HS dz for Ψ P L p,q m ppR 2d ; HSq.From Corollary 5.10 it follows that πpzqSΘpzq dz P M p 1 ,q 1 1{m and we denote this element θ.We then conclude by confirming that xV S T pzq, Θpzqy HS dz " RpV S T q " RpT q, i.e. that an arbitrary functional R P pM p,q m q 1 corresponds to an element θ P M p 1 ,q 1 1{m with the dual action defined above.Thus we have shown the reverse inclusion of pM p,q m q ˚Ă M p 1 ,q 1 1{m , and conclude pM p,q m q 1 -M p 1 ,q 1 1{m .□ Remark 5. 18.In examples so far of M p,q m operators, we have considered the rank one case, where one retrieves the familiar functions in the M p,q m spaces and their associated relations.However, the M p,q m pR d q spaces can also be related to the Schatten properties of operators, as seen by the inclusions N pL 2 ; M 1 q Ď M 1 Ă S 1 Ă HS Ă LpL 2 q Ă M 8 Ď LpL 2 ; M 8 q.
In particular we have a Gelfand triple M 1 m Ă HS Ă M 8 1{m , where the embeddings are continuous.5.5.Correspondence Principle for Operators.Finally we can give a characterisation of the spaces in terms of a coorbit structure: Theorem 5.19.For any S P A v such that }S} HS " 1, we have an isometric isomorphism M p,q m :" tT P S 1 : V S T P L p,q m pR 2d ; HSqu -tΨ P L p,q m pR 2d ; HSq : Ψ " Ψ6V S Su, under the mapping Proof.The inclusion V S pM p S q Ă tΨ P L p pR 2d ; HSq : Ψ " Ψ6V S Su follows from Lemma 5.9.It remains to show the converse.We have that Ψ6V S S " V S V S Ψ for any Ψ P L p,q m pR 2d ; HSq.Hence if Ψ " Ψ6V S S, then Ψ " V S R, where R " V S Ψ P M p,q m , since V S : L p,q m Ñ M p,q m .We recall that V S is injective on M p,q m , and the isometry property follows simply as a result of definitions of M p,q m norms for a normalised S. Hence we have the correspondence principle; tT P S 1 : V S T P L p,q m pR 2d ; HSqu -tΨ P L p,q m pR 2d ; HSq : Ψ " Ψ6V S Su, for any S P A v with }S} HS " 1. □

Atomic Decomposition
Coorbit spaces were introduced as a means of giving atomic decompositions with respect to unitary representations, and are fundamental to the field of time-frequency analysis for this reason.It is therefore natural, once one has such spaces, to consider the resulting discretisation.In particular we are interested in the discretisation of the identity for T P M p,q m and S P A v , and the g-frame condition A}T } HS ď ÿ λPΛ }S ˚πpλq ˚T } HS ď B}T } HS (16) for some lattice Λ Ă R 2d .We then proceed to consider the corresponding statement for operator modulation spaces, and interpret this as the statement that for T an operator with poor time-frequency concentration, in some M p,q m for large p, q, we can nonetheless decompose T into well localised operators in the above manner.
In [33], a similar problem was considered, of the conditions for which decompositions of functions ψ P M p m pR d q of the form ÿ λPΛ α λ pSS ˚qψ converge in a given norm.In that work the primary operators of interest were those of the form S P M 1 v pR d q b M 1 v pR d q, although as discussed in Remark 7.8 of that work, the same results hold for operators S " ř n f n bg n where tg n u n is an orthonormal system in L 2 pR 2d q, and tf n u n Ă M 1 v , with the condition With the twisted convolution structure of our coorbit spaces already in place, atomic decomposition results can be derived in an almost identical way to the function case, as presented in chapter 12 of [22], with some slight changes to accommodate the operator setting, based on the Wiener Amalgam spaces defined in Definition 2.2.We present the proofs here for completeness.Lemma 6.1.Given G P W pL 1 v pR 2d ; HSqq and F P L p,q m pR 2d ; HSq continuous functions, where m is a v-moderate weight, we have }F 6G} W pL p,q m q ď C}F } L p,q m }G} W pL 1 v q .Proof.We construct the function G s pzq " ř λPΛ S λ ¨χΩ λ pzq, where Ω λ " λ `r0, 1s 2d and S λ is a value of G in Ω λ which maximises }Gpzq} HS , which exists since G is assumed to be continuous.Then }Gpzq} HS ď }G s pzq} HS and }G} W pL 1 v q " }G s } W pL 1 v q .We then have " }F 6χ Ω 0 } W pL p,q m q }G} W pL 1 v q .We abuse notation here by taking the twisted convolution of a vector valued and scalar valued function, but we interpret F 6χ Ω 0 pzq simply as ş z´Ω 0 F pz 1 qcpz, z 1 qdz 1 .We also comment that while S λ may not be the value of G maximising F 6G, it nonetheless provides the upper bound in the first line.We consider the sequence a λ " ess sup " p}F } HS ˚χΩ 0 qpz `λq where Ω0 " r´1, 1s 2d , and }F } HS is considered a scalar valued L p,q m function.Moreover, we see that a λ χ Ω λ pzq ď p}F } HS ˚χΩ 0 qpλ`zq for z P r0, 1s 2d , where here Ω0 " r´2, 2s 2d , and so ÿ λPΛ a λ χ Ω λ pzq ď p}F } HS ˚χΩ 0 qpzq.

Finally we conclude
where we have used Young's inequality for mixed norm spaces in the last line.The claim follows.□ In the function case, V φ 0 φ 0 P W pL 1 v pR 2d qq, from which it follows that V S 0 S 0 P W pL 1 v pR 2d ; HSqq, since }V S 0 S 0 pzq} HS " |V φ 0 φ 0 pzq|.Corollary 6.2.If T P M p,q m and S P A v , then V S T P W pL p,q m pR 2d ; HSqq.
Proof.From Lemma 6.1, for any S P A v , V S 0 S P W pL 1 v pR 2d ; HSqq.By then considering V S 0 S6V S S 0 in the equation ( 12), it follows that V S S P W pL 1 v pR 2d ; HSqq again by Lemma 6.1.The corollary then follows for general T P M p,q m from Lemma 2.8 and Lemma 6.1.□ With these preliminaries the boundedness of the analysis operator follows painlessly as in the function case presented in [22]; Proposition 6.3.For S P A v , the analysis operator C S : M p,q m Ñ l p,q m pΛ, HSq, defined by C S pT q " tS ˚πpλq ˚T u λPΛ , is a bounded operator with norm }C S } ď C}V S S} W pL 1 v q , where the constant C depends only on the lattice Λ and weight vpzq Proof.By Corollary 6.2, V S S P W pL 1 v pR 2d ; HSqq.Since V S T is continuous, we have from Corollary 6.2 and Proposition 2.3 that On the other hand, we find that the synthesis operator is similarly bounded, again in the same manner as the function case of [22]: Proposition 6.4.For S P A v , the synthesis operator D S : l p,q m pΛ, HSq Ñ M p,q m , defined by D S ppT λ q λPΛ q " ÿ λPΛ πpλqST λ is a bounded operator with norm }D S } ď C}V S S} W pL 1 v q .Convergence is interpreted to be unconditional for p, q ă 8, otherwise weak*, and the constant C depends only on the lattice Λ and weight vpzq.
Proof.We are required to show that V S D S ppT λ q λPΛ q P L p,q m .By definition; }V S D S ppT λ q λPΛ qpzq} HS " } We have from Corollary 6.2 that V S S P W pL 1 v q, so we denote once more Gpzq " ř λPΛ S λ ¨ξΩ λ pzq, where S λ is the value of V S S maximising the norm over λ`r0, 1s 2d as in Lemma 6.1.We see then that the L p,q m norm is bounded (up to a constant) by the discrete l p,q m norm of the convolution of sequences s " p}S λ } HS q and t " p}T λ } HS q, and hence }V S D S ppT λ q λPΛ q} L p,q m ď C 1 }s ˚t} l p,q m ď C 2 }s} l 1 ṽ }t} l p,q m , and since }V S S} W pL 1 v q " }s} l 1 ṽ , it follows that }D S } ď C}V S S} W pL 1 v q .Unconditional convergence for p, q ă 8 follows from the boundedness of D S , since finite sequences are dense in l p,q m .For the case p " 8 or q " 8, the same fact can be used for the series xR, , for all R P M 1 v .□ Corollary 6.5.Given S, R P A v , the frame operator O S,R :" D S C R is a bounded operator on M p,q m for all 1 ď p, q ď 8 and v-moderate weights m, with operator norm }O S } ď C}V S S} W pL 1 v q }V R R} W pL 1 v q .As a final corollary, we see that Gabor g-frames for operators in A v generate equivalent norms on M p,q m .We note that while stated for general S, R, we can always consider the canonical dual frame tS ˚πpλq ˚O´1 u given a Gabor g-frame S P A v .Corollary 6.6.If S, R P A v are dual Gabor g-frames, so O S,R " I HS , then O S,R " O R,S " I M p,q m where the sum is unconditional for all 1 ď p, q ă 8, and weak* otherwise.Furthermore, there are constants A, B such that A}T } M p,q m ď }S ˚πpλq ˚T } l p,q m ď B}T } M p,q m (and similarly for R).
Remark 6.7.It would be nice to be able to decompose an operator T P HS solely in terms of α λ shifts of some window S P HS, that is, in the form T " ř λ c λ α λ pSq.In general this is impossible, since there does not exist and operator S P HS and lattice Λ such that the linear span of tα λ pSqu λPΛ is dense in HS (Proposition 7.2, [32]).This is roughly due to the fact that one must have control of both sides of the tensor product L 2 pR d q b L 2 pR d q, The Gabor g-frame condition on L 2 pR d q then gives ÿ n A}f n } The opposite direction follows by the same expansion with a rank one operator. □ The following characterisation then uses Proposition 7.14 of [33], which states that given h P L 1 v 2 pR d q, A φ h P M 1 v pR d q b M 1 v pR d q, which in particular tells us A φ h P A v .Corollary 6.10.Given h P L 1 v 2 pR 2d q satisfying (17) and some v-moderate m, the operator T P M 8 1{v belongs to M p,q m if and only if ␣ A φ h πpλq ˚T ( λPΛ P l p,q m pΛ; HSq.In a similar manner to Remark 5.2, this corollary supports the intuition of the M p,q m condition measuring the time-frequency decay in the operator sense.We often consider localisation operators with symbol h having essential support concentrated in some domain Ω, such as the characteristic function χ Ω .Hence A φ h can be seen as measuring the time frequency concentration of a function in Ω.With this intuition we can consider A φ h πpλq ˚T as measuring how much T concentrates a function to some domain Ω `λ, and thus we interpret the sum over λ as a measure of the extent to which T spreads out functions in the time-frequency plane.

Final Remarks
This paper introduces an operator STFT, a novel concept bearing potential both in theoretical settings and applications to data analysis and quantum harmonic analysis.The main results of the paper arise from representing an ensemble of data points or signals with respect to a joint timefrequency representation, which captures correlations between data points in the time-frequency plane (Example 3.5).We show that the operator STFT has many of the familiar properties of the function STFT, and in particular that the spaces produced by the operator STFT with a fixed window are reproducing kernel Hilbert spaces.The Toeplitz operators associated with such spaces are the mixed-state localisation operators, an observation, that supports the notion of the operator STFT extending the function STFT to an appropriate object for quantum harmonic analysis (Section 4).From a functional data analysis point of view, stable representations of continuous data is the ideal, and so having a reproducing structure when analysing data sets ensures stability with respect to noise and small perturbations in the incoming data.Furthermore, by extending the spaces of operators one considers, we are able to define coorbit spaces for operators.It turns out, that we thus obtain Banach spaces of operators behaving analogously to the function coorbit spaces, with regards to duality, the equivalence of window in the definition, and even the correspondence principle for the coorbit spaces (Section 5).If we interpret function coorbit spaces as those functions appropriately concentrated in a region of the time-frequency plane, we can consider the operator coorbit spaces as those operators which act on functions in a concentrated region of the time-frequency plane (Remark 5.2).These coorbit spaces of operators turn out to have the remarkable property of decomposition via Gabor g-frames, which says that given an operator in a coorbit space, we can write the operator as a sum of translations of well localised operators (Section 6).
for positive constants A, B, and almost all z P R 2d .Then for every vmoderate weight m and 1 ď p ă 8 the operator T P M 8

2. 5 . 1 .
Schatten class and nuclear operators.In this work we consider several spaces of operators.We begin by defining the trace class operators as S 1 :" tT P LpL 2 pR d qq : ÿ nPN x|T |e n , e n y ă 8u

Example 3 . 5 .
For a data operator S " ř n f n b e n , V S Spzq " ÿ n,m V fn f m pzqe n b e m .

Figure 3 .
Figure 3.The common total correlation function of both S 1 and S 2