XIANG ZHONG-QI

(College of Mathematics and Computer Science,Shangrao Normal University, Shangrao,Jiangxi,334001)

A New Characterization on g-frames in Hilbert C?-Modules

XIANG ZHONG-QI

(College of Mathematics and Computer Science,Shangrao Normal University, Shangrao,Jiangxi,334001)

Communicated by Ji You-qing

In this note,we establish a new characterization on g-frames in HilbertC?-modules from the operator-theoretic point of view,with which we provide a correction to one result recently obtained by Yao(Yao X Y.Some properties of g-frames in HilbertC?-modules(in Chinese).Acta Math.Sinica,2011,54(1):1–8.).

HilbertC?-module,g-frame,characterization

1 Introduction

The frames for a Hilbert space were introduced in the paper[1]by Duffi n and Schaef f er from 1952,when they were used to study some deep problems in nonharmonic Fourier series. The importance of frames was not realized until 1986 when Daubechieset al.[2]found a fundamental new application,to wavelet and window Fourier transform.Since then,frames have become the focus of active research,both in theory and in applications,such as the characterization of function spaces,digital signal processing,scientif i c computations,etc.

The theory of frames in Hilbert spaces was rapidly extended and various generalizations of frame concept were developed.Among them,g-frames,proposed by Sun[3],include many other generalizations of frames,e.g.,frames of subspaces(see[4]),oblique frames(see[5]), pseudo-frames(see[6])and outer frames(see[7]),etc.

The frames and g-frames for Hilbert spaces have natural analogues for HilbertC?-modules(see[8]and[9]).Although HilbertC?-modules are generalizations of Hilbert spaces,there are many essential dif f erences between them because of the complexity oftheC?-algebras involved in the HilbertC?-modules and the fact that some useful techniques available in Hilbert spaces are either absent or unknown in HilbertC?-modules.This suggests that the generalization of frame theory from Hilbert spaces to HilbertC?-modules is not a trivial task.The properties of g-frames in HilbertC?-modules were further investigated in[10]–[12].

In this paper,we study the equivalent characterization of g-frames in HilbertC?-modules. The motivation derives from an observation on a result obtained by Yao[11],which can be expressed as follows:an adjointable operator preserves g-frames in HilbertC?-modules if and only if it is invertible.However,a counterexample(see Example 3.1)shows that the“only if”part of the result is not true.In Section 3 of the present paper,we give a new characterization of g-frames in HilbertC?-modules in terms of operators(see Theorem 3.1) so that her result can be corrected(see Theorem 3.2).

2 Preliminaries

In the following we brief l y recall some def i nitions and basic properties of operators and g-frames in HilbertC?-modules.

We f i rst give some notations.Throughout this paper,the symbols J andArefer,respectively,to a f i nite or countable index set and a unitalC?-algebra.H,KandKj(for eachj∈J)are HilbertC?-modules overA(or simply,HilbertA-modules).We denote bythe set of all adjointable operators fromHtoK,andis abbreviated to.ForT∈,the notationsR(T)andN(T)are reserved respectively for the range and the null space ofT.

Def i nition 2.1For each j∈J,let Λj∈.Then we call{Λj}j∈Ja g-frame for H with respect to{Kj}j∈J,if there exist two constants C,D＞0such that

We call C,D the lower and upper g-frame bounds,respectively.The g-frame{Λj}j∈Jis said to be λ-tight if C=D=λ,and said to be Parseval if C=D=1.The sequence{Λj}j∈Jis called a g-Bessel sequence with bound D if we only require the right hand inequality of(2.1).

For each g-Bessel sequenceforHwith respect to{Kj}j∈J, we def i ne the HilbertC?-module overAassociated with{Λj}j∈Jby

and with theA-valued inner product given by

Def i nition 2.2J.

The synthesis operator U:?2({Kj}j∈J)→H is def i ned by

The adjoint operator U?:H→?2({Kj}j∈J)given by U?f={Λjf}j∈Jis called the analysis operator.By composing U and U?,we obtain the g-frame operator

The following lemmas are used in the proof of Theorems 3.1 and 3.2.

Lemma 2.1[13]Let T∈.Then the following statements are equivalent:

(1)T is surjective;

(2)T?is bounded below with respect to norm,i.e.,there is an m＞0such that‖T?f‖≥m‖f‖for all f∈K;

(3)T?is bounded below with respect to the inner product,i.e.,there is an m′＞0such that〈T?f,T?f〉≥m′〈f,f〉for all f∈K.

The arguments in Proposition 2.1 of[13]lead to an immediate consequence as follows. We include the proof for the sake of completeness.

Lemma 2.2If T∈is surjective,then TT?is an invertible operator satisfying

whereIdKis the identity operator on K.

Proof.SinceTis surjective,R(T)=Kis closed and,consequently,N(T)⊕R(T?)=HandN(T?)⊕R(T)=K.Letxbe an element inKsuch thatTT?x=0,thenT?x∈N(T)∩R(T?),meaning thatT?x=0.Notice,however,thatx∈N(T?)=(R(T))⊥=K⊥={0},we conclude thatTT?is injective.Now letzbe an arbitrary element ofK. Then there isy∈Hsuch thatz=Ty.Since there arey1∈N(T)andx∈Ksatisfyingy=y1⊕T?x,we have

which implies thatTT?is surjective.Thus,TT?is an invertible operator.Clearly,

So

We also have

Hence,

The proof is completed.

Lemma 2.3[12]For each j∈J,let Λj∈.Then{Λj}j∈Jis a g-frame for H with respect to{Kj}j∈Jif and only if the operator U def i ned by(2.3)is a well def i ned bounded operator from ?2({Kj}j∈J)onto H.

3 Main Results and Their Proofs

The following assertion is stated in[11]as Theorem 2.4.

Assertion 3.1Let{Λj∈be a g-frame for H with respect to{Kj}j∈Jwith g-frame bounds C,D and g-frame operator S,and L∈.Then{ΛjL}j∈Jis a g-frame for H with respect to{Kj}j∈Jif and only if L is invertible.In this case the lower and upper g-frame bounds for{ΛjL}j∈Jare‖L?1‖?2C and‖L‖2D,respectively.

Indeed,if{Λj∈is a g-frame forHwith respect to{Kj}j∈JandL∈is invertible,then{ΛjL}j∈Jis a g-frame forHwith respect to{Kj}j∈J. But the converse is not true in general.Although in the proof the author proved thatL?SLis invertible,it does not imply thatLis invertible onH.The reader can check the following example.

Example 3.1Letl∞be the set of all bounded complex-valued sequences.For anyu={uj}j∈N,v={vj}j∈N∈l∞,we def i ne

ThenA={l∞,‖·‖}is aC?-algebra.

LetH=C0be the set of all sequences converging to zero.For anyu,v∈Hwe def i ne

ThenHis a HilbertA-module.

it follows that

Hence,{Λj}j∈Nis a Parseval g-frame forHwith respect to{Kj}j∈N.It is obvious that the g-frame operatorSfor{Λj}j∈Nis the identity operator onH.

We def i ne a shift operator onHbyLej=ej+1,j∈N.For anyf,g∈H,since

Thus{ΛjL}j∈Nis a Parseval g-frame forHwith respect to{Kj}j∈N.Now for eachf∈H,

implying thatL?SL=IdHis invertible.Obviously,L?e1=0,soLis not invertible.

In the following we state a characterization of g-frames in HilbertC?-modules as images of an adjointable operator which is bounded below with respect to norm under a family of projections for the purpose of providing a correction to Assertion 3.1.

Theorem 3.1For each j∈J,let Λj∈(H,Kj).Then{Λj}j∈Jis a g-frame for H with respect to{Kj}j∈Jif and only if Λj=PjΓ?for all j∈J,where Γ is an adjointable operator from ?2({Kj}j∈J)onto H,and Pjis the projection on ?2({Kj}j∈J)that maps each element to its j-th component,that is,

Proof.Assume f i rst that{Λj}j∈Jis a g-frame forHwith respect to{Kj}j∈J.Then by Lemma 2.3,its synthesis operatorUis surjective.SinceΛif=Pi({Λjf}j∈J)=PiU?ffor alli∈J and allf∈H,it follows thatΛi=PiU?for alli∈J.

For the opposite implication,suppose thatΓis an adjointable operator from?2({Kj}j∈J) ontoHfor whichΛj=PjΓ?for allj∈J.Then for eachf∈H,

By Lemma 2.2 we have

which shows that{Λj}j∈Jis a g-frame with g-frame bounds‖(ΓΓ?)?1‖?1,‖?！?.The proof is completed.

Now we can correct Assertion 3.1 as follows.

Theorem 3.2Suppose that{Λj∈(H,Kj)}j∈Jis a g-frame for H with respect to{Kj}j∈Jwith g-frame bounds C,D and g-frame operator S and let L∈(H).Then {ΛjL}j∈Jis a g-frame for H with respect to{Kj}j∈Jif and only if L is bounded below with respect to norm.In this case the lower and upper g-frame bounds for{ΛjL}j∈Jare‖(L?L)?1‖?1C and‖L‖2D,respectively.

Proof.Since{Λj}j∈Jis a g-frame forHwith respect to{Kj}j∈J,by Theorem 3.1 we know that there exists a surjective operator?！??2({Kj}j∈J),H)such thatΛj=PjΓ?for eachj∈J and,ΛjL=Pj(L?Γ)?as a consequence.Again by Theorem 3.1,{ΛjL}j∈Jis a g-frame forHwith respect to{Kj}j∈Jif and only ifL?Γis surjective,which is equivalent toLis bounded below with respect to norm by Lemma 2.1.

Now letL∈(H)be bounded below with respect to norm such that{ΛjL}j∈Jis a g-frame forHwith respect to{Kj}j∈J.Denote the g-frame operator of{ΛjL}j∈JbyS′, then for eachf∈Hwe have

Combination of Lemmas 2.1 and 2.2 yields

This completes the proof.

Corollary 3.1For each j∈J,let Λj∈(H,Kj)and L∈(H).If both {ΛjL}j∈Jand{ΛjL?}j∈Jare g-frames for H with respect to{Kj}j∈J,then{Λj}j∈Jis a g-frame for H with respect to{Kj}j∈J.

Proof.Since{ΛjL}j∈Jand{ΛjL?}j∈Jare both g-frames forHwith respect to{Kj}j∈J,it follows from Theorem 3.2 that bothLandL?are bounded below respect to norm.Thus,by Lemma 2.1,Lis an invertible operator.Again by Theorem 3.2,{Λj}j∈J={(ΛjL)L?1}j∈Jis a g-frame forHwith respect to{Kj}j∈J.

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tion:46L99,42C15,46H25

A

1674-5647(2017)02-0129-06

10.13447/j.1674-5647.2017.02.04

Received date:Aug.21,2015.

Foundation item:The NSF(11271148,11561057)of China,the NSF(20151BAB201007)of Jiangxi Province, and the Science and Technology Project(GJJ151061)of Jiangxi Education Department.

E-mail address:lxsy20110927@163.com(Xiang Z Q).