A Wiener Tauberian theorem for operators and functions

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Journal of Functional Analysis

www.elsevier.com/locate/jfa

A Wiener Tauberian theorem for operators and functions

Franz Luef^∗, Eirik Skrettingland

DepartmentofMathematicsNTNUNorwegianUniversityofScienceand Technology,NO–7491Trondheim,Norway

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received10May2020 Accepted17November2020 Availableonline14December2020 CommunicatedbyK.Seip

MSC:

40E05 47G30 47B35 47B10

Keywords:

Tauberiantheorem Localizationoperators Toeplitzoperators Gaborspaces

We prove variants of Wiener’s Tauberian theorem in the framework of quantum harmonic analysis, i.e. for convolu- tionsbetweenan absolutely integrablefunctionanda trace classoperator, or of two trace classoperators. Our results includeWiener’sTauberiantheoremasaspecialcase.Appli- cationsofourTauberiantheoremsarerelatedtolocalization operators,Toeplitzoperators,isomorphismtheoremsbetween Bargmann-Fock spacesandquantization schemes withcon- sequences for Shubin’s pseudodifferential operator calculus andBorn-Jordan quantization. Based on the links between localizationoperatorsandTauberiantheorems wenotethat theanalogueof Pitt’sTauberiantheorem inour settingim- pliescompactnessresults for Toeplitz operatorsin termsof theBerezintransform.Inaddition,weextendtheresultson ToeplitzoperatorstootherreproducingkernelHilbertspaces inducedby theshort-time Fouriertransform,knownas Ga- borspaces. Finally,weestablish theequivalenceofWiener’s Tauberiantheoremandtheconditioninthecharacterization ofcompactnessoflocalizationoperatorsduetoFernándezand Galbis.

©2020TheAuthor(s).PublishedbyElsevierInc.Thisisan openaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

* Correspondingauthor.

E-mailaddresses:[email protected](F. Luef),[email protected](E. Skrettingland).

https://doi.org/10.1016/j.jfa.2020.108883

0022-1236/©2020TheAuthor(s). PublishedbyElsevierInc. ThisisanopenaccessarticleundertheCC BYlicense(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

InoperatortheoryoneviewsthespaceoftraceclassoperatorsS¹asthenoncommuta- tiveanalogueofthespaceofabsolutelyintegrablefunctionsL¹(R^d) byviewingthetrace of an operatoras the substituteof the Lebesgueintegral ofa function.Over theyears this point of viewhasled to anumber ofresults inoperatortheory where onehasex- tendedconceptsforfunctionstooperatorsinanattempttoformulateoperator-theoretic analogues of statementsabout functions. Guided by this meta-statement, Werner has proposed an operator-theoretic variant of harmonic analysis in [57], which originated from hisworkinquantum physicsandisthusreferredto as“quantum harmonicanaly- sis”.

In thispaper weestablishaversionof Wiener’sTauberiantheorem inthesetting of quantum harmonic analysis.Wiener’s Tauberiantheorem is acornerstone of harmonic analysis.Inshort,itanalysestheasymptoticpropertiesofaboundedfunctionbytesting itwith convolutionkernels.

Theorem (Wiener’s Tauberian theorem).Suppose f ∈L^∞(R^d)and h∈L¹(R^d)with a non-vanishing Fouriertransformh.Thenthefollowingimplicationholds forA∈C:if

x→∞lim(h∗f)(x) =A

R^d

h(y)dy,

then forany g∈L¹(R^d)wehave

x→∞lim(g∗f)(x) =A

R^d

g(y)dy.

Moreover, Wiener noticed that the Tauberian condition holds only for h ∈ L¹(R^d) satisfying the condition h(ω) = 0 for any ω ∈ R^d. The key step in the proof of this equivalenceisbasedonthefollowingapproximationtheorem.Forf ∈L¹(R^d) wedenote byTxf(t)=f(t−x) thetranslateoff byx∈R^d.

Theorem (Wiener’s approximation theorem).For f ∈L¹(R^d) we have that span{Txf : x∈R^d}=L¹(R^d)ifandonly if f(ω)= 0 forany ω∈R^d.

In quantum harmonic analysis one complements the convolution f ∗ g(x) =

R^df(t)g(x−t) dt of f,g ∈ L¹(R^d) with two new convolution operations: the con- volution f S of f ∈ L¹(R^d) and atrace class operatorS, and the convolution S T of two trace class operators S and T. This is achieved by replacing, for z ∈ R^2d, the translation T_zf of afunctionby the translation α_z(R) ofa bounded operatorR given by

αz(R) =π(z)Rπ(z)^∗ forz∈R^2d,

where (π(z)ψ) (t)=e^2πiω·tψ(x−t) denotesthetime-frequency shift ofψ ∈L²(R^d) by z= (x,ω)∈R^2d.

Forf ∈L¹(R^2d) andS ∈ S¹, whereS¹ denotes thetraceclassoperators,theconvo- lutionf S ∈ S¹ isthendefinedbytheBochnerintegral

f S:=

R^2d

f(z)αz(S)dz,

whichisanothertraceclassoperator.Theconvolution S T oftwooperators S,T ∈ S¹ isthefunction

S T(z) = tr(Sα_z( ˇT)) forz∈R^2d, whereTˇ=P T P,with P theparityoperatorP ψ(t)=ψ(−t).

Insummary, theconvolutions f Sand S T ariseas extensions oftheconvolution offunctions whereonereplaces eitheroneorboth L¹-functionswith traceclassopera- tors. The seminal paper [57] contains a numberof operator-theoretic versions of basic resultsfromharmonicanalysis,e.g.theRiemann-Lebesguelemma,theHausdorff-Young theorem and Wiener’sapproximation theorem.The variantof Wiener’s approximation theorem in[57] concerns translates of a trace class operator being dense in the space of trace class operators, and is established by defining an operator-theoretic Fourier transform,theFourier-WignertransformFW(S)∈L^∞(R^2d) of atraceclassoperatorS.

TheappropriateFouriertransformforfunctionsinL¹(R^2d) isthesymplecticFourier transformFσandthefollowingclassesoffunctionsandoperatorsaregoingtobecrucial inourTauberiantheoremsforquantumharmonicanalysis:

W(R^2d) :={f ∈L¹(R^2d) :Fσ(f)(z)= 0 for anyz∈R^2d}, W :={S∈ S¹:FW(S)(z)= 0 for anyz∈R^2d}.

OurfirstmainresultisageneralizationofWiener’sTauberianTheoremforfunctionson R^2d. Here Kdenotes thespace ofcompactoperators onL²(R^d) and I_L² istheidentity operator.

Theorem4.1(Tauberiantheoremforboundedfunctions).Letf ∈L^∞(R^2d),andassume that oneof thefollowing equivalentstatements holdsforsome A∈C:

(i) There issome S∈ W suchthat

f S=A·tr(S)·IL²+K forsomecompactoperatorK∈ K.

(ii) There issomea∈W(R^2d)suchthat f ∗a=A·

R^2d

a(z)dz+h

forsomeh∈C₀(R^2d).

Then bothof thefollowingstatements hold:

(1) For any T ∈ S¹,f T =A·tr(T)·I_L²+K_T forsome compactoperatorK_T ∈ K. (2) For any g∈L¹(R^2d),f∗g=A·

R^2dg(z)dz+h_g forsomeh_g∈C₀(R^2d).

We note thattheequivalence (ii) ⇐⇒ (2) isWiener’s original Tauberiantheorem.

Similarly to Wiener’s Tauberian theorem,this theorem concerns theasymptotic prop- erties ofthe operatorR when weusethe commonintuition thatasymptotic properties of an operator are properties that are invariant under compact perturbations, see [7, Chap. 3].There isalsoaversionof theprecedingtheoremforbounded operators:

Theorem 5.1 (Tauberian theorem for bounded operators).Let R ∈ L(L²), and assume that one ofthefollowingequivalent statements holdsforsomeA∈C:

(i) Thereissome S∈ W such that

R S=A·tr(S) +h forsomeh∈C₀(R^2d).

(ii) There issomea∈W(R^2d)suchthat R a=A·

R^2d

a(z)dz·I_L²+K

forsomecompactoperatorK∈ K. Then bothof thefollowingstatements hold:

(1) For any T ∈ S¹,R T =A·tr(T)+h_T forsomeh_T ∈C₀(R^2d).

(2) For any g∈L¹(R^2d),R g=A·

R^2dg(z)dz·IL²+Kg forsomecompactoperator Kg∈ K.

These Tauberian theorems have numerous applications to localization operators, Toeplitz operators and quantizationschemes. The linkto localization operators allows us to add another equivalent assumption to Theorem 4.1, formulated in terms of the

short-timeFouriertransform.Recallthattheshort-timeFouriertransformVφψofψfor thewindowφisgivenbyVφψ(z)=ψ,π(z)φ.

Proposition 4.3. Let A ∈ C. Then f ∈ L^∞(R^2d) satisfies theequivalent conditions (i) and(ii)in Theorem4.1if andonly if

(iii) Thereissomenon-zero SchwartzfunctionΦon R^2d suchthatforevery R >0

|xlim|→∞ sup

|ω|≤R|VΦ(f −A)(x, ω)|= 0.

As condition(ii) in Theorem 4.1is thecondition from Wiener’s classicalTauberian theorem, condition (iii) above, which first appeared inthe context of localization op- erators in [27], is a new characterization of the functions to which Wiener’s classical Tauberiantheoremapplies.

Tobeprecise,thelocalizationoperatorA^ϕ_f^1,ϕ2 withmaskf ∈L^∞(R^2d) andwindows ϕ1,ϕ2∈L²(R^d),isdefinedby

A^ϕ_f^1,ϕ2(ψ) =

R^2d

f(z)Vϕ1ψ(z)π(z)ϕ2 dz.

ThelinkfromlocalizationoperatorstoTheorem4.1isthenthesimplerelationA^ϕ_f^1,ϕ2 = f (ϕ2⊗ϕ1),whereϕ2⊗ϕ1(ψ)=ψ, ϕ1_L²ϕ2.Localizationoperatorsarefurtherlinked toToeplitzoperatorsonGaborspacesV_ϕ(L²) –whichcontaintheBargmann-Fockspace asaspecialcase–thisallowsthestudyofToeplitzoperators usingTheorem4.1.

The Gabor space associated with ϕ with ϕL² = 1 is Vϕ(L²) := Vϕ(L²(R^d)) ⊂ L²(R^2d).TheGaborspaceVϕ(L²) isareproducingkernelHilbertspacewithreproducing kernel

k^ϕ_z(z) =π(z)ϕ, π(z)ϕL² =Vϕ(π(z)ϕ)(z),

foranyψ∈L²(R^d).WewillshowthattheintersectionofdifferentGaborspacesistrivial wheneverthewindowsarenotscalarmultiplesof eachother. Everyf ∈L^∞(R^2d) then definesaGaborToeplitz operatorT_f^ϕ:Vϕ(L²)→Vϕ(L²) by

T_f^ϕ(V_ϕψ) =PVϕ(L²)(f·V_ϕψ),

where PVϕ(L²) : L²(R^2d)→ V_ϕ(L²) is the orthogonalprojection. It iswell-known that T_f^ϕand A^ϕ,ϕf areunitarilyequivalent.

If the window function ϕ is the Gaussian ϕ₀(x) = e^−πx2, then V_ϕ0(L²) is, up to a simple unitary transformation, the space of entire functions on C^d known as the Bargmann-Fockspace F²(C^d).For everyF ∈L^∞(C^d) onedefines the Bargmann-Fock Toeplitzoperator T_F^F2 onF²(C^d) by

T_F^F2(H) =PF²(F·H)

for any H ∈ F²(C^d). One has that if f ∈ L^∞(R^2d) and F ∈ L^∞(C^d) are related by F(x+iω)=f(x,−ω) thefollowingoperators areunitarilyequivalent:

(1) ThelocalizationoperatorA^ϕf^0,ϕ0 :L²(R^d)→L²(R^d).

(2) TheGaborToeplitz operatorT_f^ϕ0 :Vϕ0(L²)→Vϕ0(L²).

(3) TheBargmann-FockToeplitzoperatorT_F^F2 :F²(C^d)→ F²(C^d).

Since A^ϕ_f^0,ϕ0 =f (ϕ0⊗ϕ0),the equivalences aboveallow us to translate statements from convolutions of operators to Toeplitz operators. One of the results we translate to Toeplitz operators follows by noting thatthe Tauberiantheorems concern compact perturbations ofascaling oftheidentity, i.e. operatorsA·I_L²+K for0=A∈C and K∈ K.Inspiredbythis–withoutusingtheTauberiantheoremitself–weapplyRiesz’

theory of suchoperators to obtain sufficientconditions forlocalizationoperators to be isomorphisms:

Proposition 4.10.Let0=M ∈R, a∈ L^∞(R^2d) andΔ ⊂R^2d a set of finite Lebesgue measure. Assumethat thefollowingassumptionshold:

(i) a(z)≥ −M fora.e.z∈R^2d, (ii) a(z)>−M forz /∈Δ,

(iii) a satisfiesassumption(i)or(ii)inTheorem4.1 withA= 0.

Letf =M+a.ThenA^ϕ,ϕf isan isomorphismon L²(R^d)forany 0=ϕ∈L²(R^d).

Wetranslatethese resultstothe polyanalyticBargmann-FockspaceFn²(C^d) forn∈ N^d –inparticularF0²(C^d) istheBargmann-FockspaceF²(C^d).

Proposition 4.12.

(1) IfΩ⊂C^dsatisfiesthatΩ^chasfiniteLebesguemeasure,thenT^F

2

χΩnisanisomorphism on Fn²(C^d).

(2) There isa real-valued,continuous F ∈L^∞(C^d) suchthat lim_|z|→∞|F(z)| doesnot exist, yetT_F^Fn2 isan isomorphismon Fn²(C^d).

Another class of our results concerns the Berezin transform. For the Gabor space Vϕ(L²) wecanexpresstheBerezintransformB^ϕ:Vϕ(L²)→L^∞(R^2d) asaconvolution of operators.Inparticular,theBerezin transformoftheGabor ToeplitzoperatorT_f^ϕis simplyaconvolutionof functions:

B^ϕT_f^ϕ(z) =

f∗ |Vϕϕ|² (z).

Pitt’sclassical theoremgives aconditiononf ∈L^∞(R^2d) thatensuresthatf ∗g ∈ C0(R^2d) for g ∈ W(R^2d) implies f ∈ C0(R^2d). In particular, this holds for uniformly continuousf.Anaturalanalogueofuniformlycontinuousfunctionsforoperatorsisthe set

C1:={R∈ L(L²) :z→α_z(R) is continuous fromR^2d to L(L²)},

see[12,57].Wernerhasobtainedthefollowingresultin[57] whichinlightofourTaube- riantheoremisananalogueofPitt’stheoremforoperators.

Theorem5.2. LetR∈ C1.The followingareequivalent.

• R∈ K.

• R S∈C0(R^2d)forsome S∈ W.

• R f ∈ Kforsome f ∈W(R^2d).

Fulsche[32] hasrecentlynotedthattheprecedingtheoremimpliesaresultin[11] for theBargmann-Fock space. Weshow thatthe resultholds for any Gabor space V_ϕ(L²) undercertainconditionsonϕ.Wewouldlike tostressthatitisaPitt-typetheoremfor theTauberiantheorem foroperators.

Theorem5.4. Letϕ∈L²(R^d)withϕL² = 1satisfy that Vϕϕhas nozeros,andletT^ϕ betheBanachalgebrageneratedbyToeplitzoperatorsT_f^ϕ⊂ L(Vϕ(L²))forf ∈L^∞(R^2d).

Thenthefollowingareequivalent forT˜∈ T^ϕ.

• ˜T isacompactoperatoronV_ϕ(L²).

• B^ϕT˜∈C0(R^2d).

Furthermore, if T˜=T_f^ϕ for some slowly oscillating f ∈L^∞(R^2d), then theconditions above areequivalent tolim_|z|→∞|f(z)|= 0.

Examples of ϕ satisfying that V_ϕϕ has no zeros were recently investigated in [37], forexampletheone-sidedexponential.Hencetheseϕ’sgivedifferentreproducingkernel Hilbert spaces V_ϕ(L²) such that Toeplitz operators are compact if and only if their Berezintransformvanishesatinfinity.

The main result in [11] follows in particular, as shown in [32]. We have added a statementonslowlyoscillatingfunctionsthatfollows fromtheoriginalversionof Pitt’s theorem.

Theorem5.5(Bauer,Isralowitz). LetT^F2betheBanachalgebrageneratedbytheToeplitz operatorsT_F^F2 forF ∈L^∞(C^d).The followingareequivalent forT˜∈ T^F2.

• ˜T isacompactoperatoronF²(C^d).

• B^F2T˜∈C0(C^d).

IfT˜=T_F^F2 foraslowlyoscillatingF ∈L^∞(C^d),thentheconditionsaboveareequivalent to lim_|z|→∞F(z)= 0.

As aconsequencewestateacompactnessresultforToeplitzoperators.

Corollary 5.5.1.A Toeplitz operator T_F^F2 for F ∈ L^∞(C^d) is a compact operator on F²(C^d) ifandonly if

f ∗ |V_ϕ0ϕ₀|²∈C₀(R^2d),

where f(x,ω)=F(x−iω)forx,ω∈R^d and|Vϕ0ϕ0(z)|²=e^−π|z|2.

Finally,Theorem5.2givesasimpleconditionforcompactnessoflocalizationoperators intermsoftheGaussianϕ0.

Proposition5.6.Letf ∈L^∞(R^2d)andψ1,ψ2∈L²(R^d).ThelocalizationoperatorA^ψ_f^1,ψ2 is compactifandonly if

f ∗(Vϕ0ψ2Vϕ0ψ1)∈C0(R^2d).

Finally we recall from [51] that any R ∈ L(L²) defines aquantizationscheme given by f →f R forf ∈L¹(R^2d) and atime-frequency distributionQR,given bysending ψ∈L²(R^d) toQ_R(ψ)(z)= (ψ⊗ψ)R(z) forˇ z∈R^2d.ThedistributionQ_RisofCohen’s class since we have QR(ψ) = aRˇ ∗W(ψ,ψ), where aRˇ is the Weyl symbol of Rˇ and W(ψ,ψ) theWignerdistributionofψ.

In the final section we deduce a statement relating compactness properties of the quantizationscheme off →f Rtopropertiesof QR(ψ).

Proposition 6.1. LetR∈ L(L²).The followingareequivalent.

(i) QR(ϕ)∈C0(R^2d) forsomeϕ∈L²(R^d) suchthatVϕϕhas nozeros.

(ii) g R∈ Kforsomeg∈W(R^2d).

(iii) QR(ψ)∈C0(R^2d)forallψ∈L²(R^d).

(iv) f R∈ Kforallf ∈L¹(R^2d).

HenceifonetakestheGaussianϕ0for(i),thencheckingifQR(ϕ0)∈C0(R^2d) provides a simpletest forchecking whetherConditions(iii) and (iv) hold. Weapply this result to Shubin’sτ-quantizationscheme andBorn-Jordanquantization.

1.1. Notations andconventions

FortopologicalvectorspacesX,Y,wedenotebyL(X,Y) thesetofcontinuous,linear operators from X to Y. If X = Y we write L(X) = L(X,X). The space of compact operators on L²(R^d) is denoted by K. For 1 ≤p <∞ we let S^p denote the Schatten p-classof compactoperatorswithsingularvaluesin^p,andweusetheconventionthat S^∞=L(L²).Inparticular,S¹denotesthespaceoftraceclassoperatorsonL²(R^d),and thetraceofatraceclass operatorT ∈ S¹ is denotedbytr(T). Also,S² is thespace of Hilbert-Schmidtoperators,whichformaHilbertspacewithrespecttotheinnerproduct S, T_S² = tr(ST^∗).

GivenatopologicalvectorspaceX anditscontinuousdualX,theactionofx^∗∈X ony∈xisdenotedbyx^∗, yX,X.ToagreewiththeHilbertspaceinnerproductweuse theconventionthatthedualitybracketislinearinthefirstcoordinateandantilinearin thesecondcoordinate.TheSchwartzfunctionsonR^d aredenotedby S(R^d).

TheEuclidean norm on R^d or C^d will be denoted by |· |. For Ω⊂R^d, χ_Ω denotes thecharacteristic functionof Ω.As usual, C₀(R^d) denotesthecontinuous functionson R^d vanishingatinfinity,andweuseL⁰(R^d) todenotethespaceofmeasurable,bounded functionsf onR^d suchthatlim_|z|→∞f(z)= 0,i.e. forevery>0 thereis R >0 such that |f(z)| < for a.e. |z| > R. We will refer to L^p-spaces on R^d,R^2d and C^d, and sometimeswewill omitexplicit referenceto theunderlyingspace whenit isclear from thecontext,forinstancebywritingL(L²) forL(L²(R^d)).Inallstatements,measurability and“almost everywhere”propertieswill referto Lebesguemeasure.

2. Preliminaries

2.1. Conceptsfrom time-frequencyanalysis

Themathematicaltheoryoftime-frequencyanalysiswillprovidethesetupandmany ofthetoolsweuseinthispaper.Wethereforeintroducethetime-frequencyshiftsπ(z)∈ L(L²) forz= (x,ω)∈R^2d,givenby

(π(z)ψ) (t) =e^2πiω·tψ(t−x) forψ∈L²(R^d).

Thetime-frequencyshiftπ(z) isclearlygivenas acompositionπ(z)=MωTx ofamod- ulation operator Mωψ(t) = e^2πiω·tψ(t) and atranslation operator Txψ(t) = ψ(t−x).

Given ψ,φ∈L²(R^d), theshort-timeFourier transformV_φψ ofψ withwindowφis the functiononR^2d definedby

Vφψ(z) =ψ, π(z)φ_L² forz∈R^2d.

Theshort-timeFouriertransformsatisfies theimportantorthogonalityrelation

R^2d

Vφ1ψ1(z)Vφ2ψ2(z)dz=ψ1, ψ2L²φ2, φ1L², (1)

see [30,35], sometimescalled Moyal’sidentity. Throughoutthis paperwe willuseϕ0 to denote thenormalizedGaussian

ϕ₀(t) = 2^d/4e^−πt2 fort∈R^d,

and wewill oftenreferto itsshort-timeFouriertransform, whichby[35, Lem.1.5.2] is given by

Vϕ0ϕ0(z) =e^−πix·ωe^−π|z|2/2 forz= (x, ω); (2) thereadershouldnotealreadyat thispointthatV_ϕ0ϕ₀ hasnozeros.

2.1.1. Wignerfunctions andtheWeyltransform

Given φ,ψ∈L²(R^d),aclose relativeof theshort-timeFouriertransform Vφψ is the cross-Wigner distributionW(ψ,φ) definedby

W(ψ, φ)(x, ω) =

R^d

ψ(x+t/2)φ(x−t/2)e^−2πiω·tdt for (x, ω)∈R^2d.

The cross-Wigner distributionis themain tool neededto introduce theWeyl trans- form,whichassociatestoanyf ∈S(R^2d) anoperatorLf ∈ L(S(R^d),S(R^d)) defined byrequiring

Lf(ψ), φ_S(R^d),S(R^d)=f, W(φ, ψ)_S(R^2d),S(R^2d) for allφ, ψ∈S(R^d). (3) BytheSchwartzkerneltheorem[41],anyS∈ L(S(R^d), S(R^d)) istheWeyltransform L_f forsomeuniquef ∈S(R^2d).We denotethis f bya_S, andcall itthe Weylsymbol of S.Inotherwords, S=L_aS. Notethatthereisnorelationshipbetweenboundedness ofthefunctionf and boundednessoftheoperatorLf onL²(R^d):thereisf ∈L^∞(R^2d) suchthatLf ∈ L/ (L²),and thereisS∈ L(L²) suchthataS ∈/L^∞(R^2d).SeeRemark20 forexamples.

Example2.1(Rank-oneoperators).Givenψ,φ∈L²(R^d),therank-oneoperatorψ⊗φ∈ L(L²) isdefinedby

(ψ⊗φ)(ξ) =ξ, φ_L²ψ forξ∈L²(R^d).

It iswell-knownthattheWeylsymbolofψ⊗φisW(ψ,φ).

2.1.2. Localization operators

For a mask f ∈ L^∞(R^2d) and a pair of windows ϕ1,ϕ2 ∈ L²(R^d), we define the localization operator A^ϕ_f^1,ϕ2(ψ)∈ L(L²) by

A^ϕf^1,ϕ2(ψ) =

R^2d

f(z)V_ϕ1ψ(z)π(z)ϕ₂ dz,

wheretheintegralisinterpretedweaklyinthesensethatwerequire A^ϕ_f^1,ϕ2(ψ), φ

L²(R^d)= f, Vϕ2φVϕ1ψ

L²(R^2d) for anyψ, φ∈L²(R^d). (4) Itiswell-knownthatA^ϕf^1,ϕ2isboundedonL²(R^d) forf ∈L^∞(R^2d) andϕ1,ϕ2∈L²(R^d) [20],butonemayalso definelocalizationoperators forotherBanachfunctionspacesof masksf andwindowsϕ₁,ϕ₂ byinterpretingthebracketsin(4) asdualitybrackets, see [20]. We postpone this discussion until we have amore suitable framework, which we nowintroduce.

2.2. Quantumharmonic analysis: convolutionsofoperators andfunctions

In this section we introduce the quantum harmonic analysis developed by Werner in[57], the main concepts of which are convolutions of operators and functionsand a Fouriertransformofoperators.Foramoredetailedintroductioninourterminologywe referto [50]. Given any z ∈R^2d and anoperatorR ∈ L(L²), wedefine the translation α_z(R) ofRbyz tobetheoperator

α_z(R) =π(z)Rπ(z)^∗. Atthelevel ofWeylsymbols,wehavethat

α_z(R) =L_Tz(aR),

henceαz correspondstoatranslationoftheWeylsymbol.Forf ∈L¹(R^2d) andS ∈ S¹ wethendefinetheconvolution f S∈ S¹ bytheBochnerintegral

f S :=S f:=

R^2d

f(z)αz(S)dz. (5)

Hencetheconvolution ofafunctionwith anoperatorisanew operator.Theconvo- lutionS T oftwo operatorsS,T ∈ S¹ isthefunction

S T(z) = tr(Sαz( ˇT)) forz∈R^2d. (6)

Here Tˇ=P T P,withP theparityoperatorP ψ(t)=ψ(−t).ThenS T ∈L¹(R^2d) with

R^2dS T(z)dz= tr(S)tr(T) and S T =T S[57].Takingconvolutions withafixed operatororfunctioniseasily seentobe alinearmap.

One of the most important properties of the convolutions (5) and (6) is that they interact nicelywitheachotherandwiththeusualconvolution f∗g(x)=

R^df(t)g(x− t)dtoffunctions,asismoststrikinglyshownbytheirassociativity [50,57].

Proposition2.1. Theconvolutions(5)and(6)areassociative.Writtenoutindetail,this meansthat forS,T,R∈ S¹ andf,g∈L¹(R^2d)wehave

(R S) T =R (S T) f∗(R S) = (f R) T

(f∗g) R=f (g R).

Remark 1.Special casesof this associativity haveappearedseveral times inthelitera- ture, typicallywith lesstransparentformulationsand proofsthanthose allowed bythe convolution formalism.Seeforinstance [27,Prop.3.10].

TheconvolutionsalsohaveaninterestinginterpretationintermsoftheWeylsymbol, as wehavethat

S T(z) =aS∗aT(z) (7)

af S(z) =f∗aS(z).

Asisshownindetailin[50],onecanextendthedomainsoftheconvolutionsbyduality.

Forinstance,theconvolutionf S ∈ L(L²) ofS∈ S¹ andf ∈L^∞(R^2d) isdefinedby f S, T_L(L²_),S¹ =

f,Sˇ^∗ T

L^∞,L¹.

Combining this with a complex interpolation argument gives a version of Young’s in- equality [50,57].RecallourconventionthatS^∞=L(L²).

Proposition 2.2(Young’sinequality). Let1≤p,q,r≤ ∞be suchthat _p¹+¹_q = 1+¹_r.If f ∈L^p(R^2d),S ∈ S^p andT ∈ S^q,then f T ∈ S^r andS T ∈L^r(R^2d) may bedefined and satisfythenormestimates

f TS^r ≤ fL^pTS^q, S TL^r ≤ SS^pTS^q.

Remark2.ItisworthnotingthatifS∈ S¹ and T ∈ L(L²),thenS T isstillgiven by (6),whichcanbeinterpretedpointwise,sothatS T isacontinuous,boundedfunction.

Young’s inequalityabove shows thatthe convolutions interact ina predictable way with L^p(R^2d) and S^q. We now show that the same is true for functions vanishing at infinity and compact operators. Recall that L⁰(R^2d) denotes the Banach subspace of L^∞(R^2d) consistingof f ∈L^∞(R^2d) thatvanish atinfinity.The following resultshows that convolutions with traceclass operators interchange L⁰(R^2d) and K, which is the basisforourmaintheorems.Theseresultsareknown,inparticularwementionthatpart (ii) wasprovedforrank-one operatorsS in[14] usingessentiallythesameproof.

Lemma2.3. LetR∈ K andf ∈L⁰(R^2d).If S∈ S¹,then (i) R S∈C₀(R^2d),

(ii) f S∈ K,

andif a∈L¹(R^2d)then (iii) R a∈ K,

(iv) f∗a∈C0(R^2d).

Proof. Part (i) is [50, Prop. 4.6]. For (ii) and (iv), note thatany f ∈ L⁰(R^2d) is the limitinthenorm topologyofL^∞(R^2d) ofasequenceofcompactlysupportedfunctions fn – simply pick fn = f ·χBn(0), where Bn(0) = {z ∈ R^2d : |z| < n}. Clearly fn ∈ L¹(R^2d), hencefn S ∈ S¹ ⊂ K. Wetherefore havebyYoung’s inequality(recallthat S^∞=L(L²)):

f S−fn SL(L²)=(f−fn) SL(L²)≤ f−fnL^∞SS¹→0 asn→ ∞, so f S is the limitin theoperator norm of compact operators,hence itself compact.

Similarly,fn∗a∈C0(R^2d) andfn∗aconvergesuniformlytof∗abyYoung’sinequality (f −f_n)∗aL^∞ ≤ f −f_nL^∞aL¹, so that f ∗a ∈ C₀(R^2d). Finally, (iii) follows bynoting thatany R ∈ Kis the limitinthe operatornorm of asequence R_n ∈ S¹ of finite-rankoperators.ThenR_n a∈ S¹iscompact,soitfollowsby(R−R_n) aL(L²)≤ R−RnL(L²)aL¹ thatR aisthelimitintheoperatornormofasequenceofcompact operators,henceitselfcompact.

Remark3.IncombinationwithProposition2.2andthefactthatS^p⊂ Kforp<∞,we seethatL^p(R^2d)S¹⊂ K forp= 0 and 1≤p<∞.

Finally, the convolutions preserve identity elements [57, Prop. 3.2 (3)]. Here I_L² ∈ L(L²) istheidentityoperatorand1∈L^∞(R^2d) isgivenby1(z)=z.

Lemma2.4. LetS∈ S¹ and f ∈L¹(R^2d).Then S IL² = tr(S)·1,

S 1 = tr(S)·IL², f I_L² =

R^2d

f(z) dz·I_L²,

f ∗1 =

R^2d

f(z) dz·1.

2.2.1. Fouriertransformsof functionsandoperators

As our Fourier transform of functions on R^2d we will use the symplectic Fourier transformFσ,given, forf ∈L¹(R^2d),by

Fσf(z) =

R^2d

f(z)e^{−2πiσ(z,z)}dz forz∈R^2d,

where σisthestandardsymplecticformσ((x1,ω1),(x2,ω2))=ω1·x2−ω2·x1.Clearly Fσ isrelatedto theusualFouriertransformf(z)=

R^2df(z)e^−2πiz·z dz by Fσ(f)(x, ω) =f(ω,−x),

so Fσ shares mostproperties withf: it extendsto aunitary operatoron L²(R^2d) and to abijection on S(R^2d) –see[21]. Inaddition,Fσ isitsowninverse:Fσ◦ Fσ=IL².

We willalso useaFouriertransformof operators,namely theFourier-Wignertrans- form FW introduced by Werner [57] (Werner calls it the Fourier-Weyl transform, our usageof Fourier-Wigneragrees with[30]).WhenS∈ S¹,FW(S) isthefunction

FW(S)(z) =e^−πix·ωtr(π(−z)S) forz= (x, ω)∈R^2d. (8) As is shownin [51,57], FW extends to a unitary mapping FW : S² → L²(R^2d) and a bijection onto S(R^2d) fromL(S(R^d),S(R^d)).

TheFouriertransformsinteractintheexpected waywithconvolutions [57]:ifS,T ∈ S¹and f ∈L¹(R^2d),then

Fσ(S T) =FW(S)· FW(T), (9) FW(f S) =Fσ(f)· FW(S). (10) WemayalsoconnectFW andFσ bytheWeyltransform.Infact,wehaveby[51,Prop.

3.16] that

FW(Lf) =Fσ(f) forf ∈S(R^2d). (11) A main concern for this paper will be functions and operators satisfying that the appropriate Fourier transform never vanishes. Following the notation of [49] for the functioncase,weintroducethefollowingnotation:

W(R^2d) :={f ∈L¹(R^2d) :Fσ(f)(z)= 0 for anyz∈R^2d}, W :={S∈ S¹:FW(S)(z)= 0 for anyz∈R^2d}.

Thekeytoolforproving theTauberiantheorem foroperators isthefollowinggener- alizationofWiener’s approximationtheorem,originallyprovedbyWerner[57].Seealso [47,50] formoregeneralstatements.

Theorem2.5(Werner).LetS∈ S¹.The followingare equivalent.

(1) Thelinear spanof thetranslates{αz(S)}z∈R^2d isdense inS¹. (2) S ∈ W.

(3) Theset L¹(R^2d) S={f S :f ∈L¹(R^2d)}isdense inS¹. (4) ThemapT →S T isinjectivefrom L(L²)toL^∞(R^2d).

(5) Theset S¹ S={T S:T ∈ S¹}isdensein L¹(R^2d).

(6) Themapf →f S isinjectivefrom L^∞(R^2d)toL(L²).

2.2.2. The special caseofrank-one operators

WhenS∈ S¹isarank-oneoperatorψ⊗φforψ,φ∈L²(R^d),thenmanyoftheconcepts introducedabovearefamiliarconceptsfromtime-frequencyanalysis.Firstwenotethat by[50,Thm.5.1],localizationoperatorsA^ϕf^1,ϕ2 canbedescribedas convolutionsby

A^ϕf^1,ϕ2 =f (ϕ2⊗ϕ1). (12) Other convolutions and Fourier-Wigner transforms of rank-one operators are sum- marized inthe next lemma. See[50, Thm.5.1 and Lem. 6.1] for proofs. Here ϕ(t)ˇ :=

(P ϕ)(t)=ϕ(−t).

Lemma2.6. Letϕ1,ϕ2,ξ1,ξ2∈L²(R^d)andS∈ L(L²).Then,for(x,ω)∈R^2d, (1) FW(ϕ₁⊗ϕ₂)(x,ω)=e^iπx·ωV_ϕ2ϕ₁(x,ω).

(2) S (ϕ1⊗ϕ2)(z)=Sπ(z) ˇϕ1, π(z) ˇϕ2L².

(3) (ξ₁⊗ξ₂)( ˇϕ₁⊗ϕˇ₂)(x,ω)=V_ϕ2ξ₁(x,ω)V_ϕ1ξ₂(x, ω).

Inparticular, forξ,ϕ∈L²(R^d)

(ξ⊗ξ)( ˇϕ⊗ϕ)(z) =ˇ |V_ϕξ(z)|².

Example 2.2 (Standard Gaussian).By (2), FW(ϕ₀⊗ϕ₀)(z) =e^−π|z|2/2. We point out thissimplecaseasitshowsthatϕ0⊗ϕ0∈ W. Inparticular, W isnon-empty.

3. ToeplitzoperatorsandBerezintransforms

InthissectionwewillintroducesomefamiliesofreproducingkernelHilbertspacesand thecorrespondingToeplitzoperatorsandBerezintransforms.Wewillrelatethesespaces andoperators totheconvolutions introducedinSection2.2,whichwilllaterallowusto deduce results for reproducing kernel Hilbert spaces from the main results this paper.

ByfarthemoststudiedofthespacesweconsideristheBargmann-FockspaceF²(C^d), and we will later investigate whether somewell-known result for F²(C^d) can hold for other ofthereproducingkernelHilbertspacesweconsider.

3.1. Gaborspaces Vϕ(L²)

Letϕ∈L²(R^d) withϕL²= 1.By(1),theshort-timeFouriertransform Vϕ:L²(R^d)→L²(R^2d)

is anisometry, andoneeasily confirmsthatitsadjointoperatoris V_ϕ^∗F =

R^2d

F(z)π(z)ϕ dz forF ∈L²(R^2d),

where the vector-valued integral is interpreted in a weak sense, see [35, Sec. 3.2] for details. The Gabor space associated with ϕ is then the image Vϕ(L²(R^d)) ⊂L²(R^2d), whichwe denotebyVϕ(L²) forbrevity.Onecanshowusing(1) that

V_ϕ^∗Vϕ=I_L²_(R^d₎,

VϕV_ϕ^∗=PVϕ(L²), (13) where PVϕ(L²) denotestheorthogonal projectionontothesubspaceVϕ(L²) ofL²(R^2d).

ThismeansthatVϕisaunitaryoperatorfromL²(R^d) toVϕ(L²),withinverseV_ϕ^∗|Vϕ(L²). By writing outthe operators in(13) one deduces that V_ϕ(L²) is areproducing kernel Hilbertspacewithreproducingkernel

k^ϕ_z(z) =π(z)ϕ, π(z)ϕL² =V_ϕ(π(z)ϕ)(z), (14) meaning thatwehavethereproducingformula

Vϕψ(z) =Vϕψ, k_z^ϕ_L²(R^2d)

for any ψ ∈L²(R^d). Every f ∈ L^∞(R^2d) then defines aGabor Toeplitz operatorT_f^ϕ : Vϕ(L²)→Vϕ(L²) by

T_f^ϕ(Vϕψ) =PVϕ(L²)(f·Vϕψ).