# Controlled Frame for Operator in Hilbert c∗-Modules

Authors: Hatim Labrigui; Abdeslam Touri; Mohamed Rossafi; Samir Kabbaj
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

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## Abstract

In this study, we will introduce a new concept, which is a controlledK-operator frame for the space of all adjointable operators on a Hilbert A-module ℋ which denoted EndA∗ℋ, where A is a C∗-algebra. Also, we establish some results of the controlled K-operator frame in EndA∗ℋ. The presented results are new and of interest for people working in this area. Some illustrative examples are provided to advocate the usability of our results.


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## Body

## 1. Introduction

Frame theory has a great revolution for recent years; this theory has several properties applicable in many fields of mathematics and engineering and plays a significant role in signal and image processing, which leads to many applications in informatics, medicine, and probability. Frame theory has been extended from Hilbert spaces to HilbertC∗-modules and has begun to be study widely and deeply. The basic idea was to consider module over C∗-algebra instead of linear spaces and to allow the inner product to take values in the C∗-algebra.The concept of frames in Hilbert spaces has been introduced by Duffin and Schaeffer [1] in 1952 to study some deep problems in nonharmonic Fourier series. After the fundamental paper [2] by Gabor, frame theory began to be widely used, particularly, in the more specialized context of wavelet frames and Gabor frames [3]. Frames have been used in signal processing, image processing, data compression, and sampling theory.In 2000, Frank and Larson [4] have extended the theory for the elements of C∗-algebras and Hilbert C∗-modules. Eventually, frames with C∗-valued bounds in Hilbert C∗-modules have been considered in [5].The theory of frames has been generalized rapidly and there are various generalizations of frames in Hilbert spaces and HilbertC∗-modules.In 2012, Gavruta [6] introduced the notion of K-frames in Hilbert space to study the atomic systems with respect to a bounded linear operator K.Controlled frames in Hilbert spaces have been introduced in 2010 by Balazs et al. [7] to improve the numerical efficiency of iterative algorithms for inverting the frame operator, while the notion of controlled K-frames in Hilbert spaces was introduced in 2015 by Nouri et al. [8].Controlled frames inC∗-modules were introduced by Kouchi and Rahimi [9], and the authors showed that they share many useful properties with their corresponding notions in a Hilbert space.The notion ofK-operator frame for EndA∗ℋ, which is a generalization of K-frames in Hilbert C∗-modules, is studied by Rossafi and Kabbaj [10].Motivated by the above literature, we introduce the notion of controlledK-operator frame for Hilbert C∗-modules which is a generalization of K-operator frame for EndA∗ℋ.


## 2. Preliminaries

In the following, we briefly recall the definitions and basic properties ofC∗-algebra and Hilbert A-modules. Our references for C∗-algebras are [11, 12].For aC∗-algebra A, if a∈A is positive, we write a≥0 and A+ denotes the set of all positive elements of A.Definition 1 (see [13]).
LetA be a unital C∗-algebra and ℋ be a left A-module, such that the linear structures of A and ℋ are compatible. ℋ is a pre-Hilbert A-module if ℋ is equipped with an A-valued inner product .,.A:ℋ×ℋ⟶A, such that is it is sesquilinear and positive definite and respects the module action. In other words,(i)
x,xA≥0, for all x∈ℋ, and x,xA=0 if and only if x=0(ii)
ax+y,zA=ax,zA+y,zA, for all a∈A and x,y,z∈ℋ(iii)
x,yA=y,xA∗, for all x,y∈ℋ
Forx∈ℋ, we define the norm of x by x=x,xAA1/2. If ℋ is complete with ., it is called a Hilbert A-module or a Hilbert C∗-modules over A.
For everya in C∗-algebra A, we have aA=a∗a1/2 and the A-valued norm on ℋ is defined by x=x,xA1/2, for all x∈ℋ.
Letℋ and K be two Hilbert A-modules; a map T:ℋ⟶K is said to be adjointable if there exists a map T∗:K⟶ℋ such that Tx,yA=x,T∗yA for all x∈ℋ and y∈K.
We reserve the notationEndA∗ℋ,K for the set of all adjointable operators from ℋ to K and EndA∗ℋ,ℋ is abbreviated to EndA∗ℋ.
RT denotes the range of the operator T∈EndA∗ℋ,ℋ, and the set of all adjointable, positive, and invertible operators with bounded inverse will be denoted by GL+ℋ.
The following lemmas will be used to prove our main results.Lemma 1 (see [14]).
Letℋ and K be two Hilbert A-modules and T∈EndA∗ℋ,K. The following statements are equivalent:(i)
Tis surjective.(ii)
T∗is bounded below with respect to norm, i.e., there ism>0such that(1)T∗x≥mxx∈K.(iii)
T∗is bounded below with respect to the inner product, i.e., there ism′>0such that(2)T∗x,T∗xA≥m′x,xAx∈K.Lemma 2 (see [13]).
Letℋ be a Hilbert A-module and T∈EndA∗ℋ; then, we have, for all x∈ℋ,(3)Tx,TxA≤T2x,xA.Theorem 1 (see [15]).
Letℋ be a Hilbert A-module over a C∗-algebra A and let T,S∈EndA∗ℋ. If RS is closed, then the following statements are equivalent:(1)
RT⊆RS(2)
TT∗≤λ2SS∗, for someλ≥0(3)
There existsQ∈EndA∗ℋsuch thatT=SQLemma 3 (see [16]).
Letℋ be a Hilbert A-module. If Q∈EndA∗ℋ is an invertible A-linear map, then, for all z∈ℋ⊗K, we have(4)Q∗−1−1.z≤Q∗⊗Iz≤Q.z.Lemma 4 (see [5]).
Ifφ:A⟶ℬ is a ∗-homomorphism between C∗-algebras, then φ is increasing, that is, if a≤b, then φa≤φb.


## 3. ControlledK-Operator Frame for EndA∗ℋ

We begin this section with the following definitions.Definition 2 (see [10]).
LetK∈EndA∗ℋ. A sequence of adjointable operators Tii∈I on a Hilbert A-module ℋ over a unital C∗-algebra is called a K-operator frame for EndA∗ℋ if there exist two positive constants A,B>0 such that(5)AK∗x,K∗xA≤∑i∈ITix,TixA≤Bx,xA,x∈ℋ.
The numbersA and B are called lower and upper bound of the operator frame, respectively.Definition 3 (see [17]).
LetC,C′∈GL+ℋ. A sequence of adjointable operators Tii∈I on a Hilbert A-module ℋ over a unital C∗-algebra is said to be a C,C′-controlled operator frame for EndA∗ℋ if there exist two positive constants A,B>0 such that, for all x∈ℋ, we have(6)Ax,xA≤∑i∈ITiCx,TiC′xA≤Bx,xA.
The numbersA and B are called lower and upper bounds of the C,C′-controlled operator frame, respectively.
Now, we define the notion of controlledK-operator frame for EndA∗ℋ.Definition 4.
LetK∈EndA∗ℋ and C,C′∈GL+ℋ. A sequence of adjointable operators Tii∈I on a Hilbert A-module ℋ over a unital C∗-algebra A is said to be a C,C′-controlled K-operator frame for EndA∗ℋ if there exist two positive constants A,B>0 such that(7)AK∗x,K∗xA≤∑i∈ITiCx,TiC′xA≤Bx,xA,x∈ℋ.
The numbersA and B are called lower and upper bound of the C,C′-controlled K-operator frame, respectively.
IfA=B=λ, the C,C′-controlled K-operator frame is called λ-tight.
IfA=B=1, it is called a Parseval C,C′-controlled K-operator frame.
If only upper inequality of (7) holds, then Tii∈i is called a C,C′-controlled K-operator Bessel sequence for EndA∗ℋ.Example 3.4.
LetC∈GL+ℋ, K∈EndA∗ℋ, and xii∈I be a C-controlled K-frame for ℋ.
LetΓii∈I⊂EndA∗ℋ such that(8)Γix=x,xiAei,for alli∈Iandx∈ℋ,where ei,ejA=δij1A.
From the definition ofxii∈I, there exist two constants A,B>0 such that, for all x∈ℋ, we have(9)AC1/2K∗x,C1/2K∗xA≤∑i∈Ix,xiACxi,xA≤Bx,xA.
Hence,(10)AC1/2K∗x,C1/2K∗xA≤∑i∈Ix,xiAei,eiACxi,xA≤Bx,xA,so(11)AC1/2K∗x,C1/2K∗xA≤∑i∈Ix,xiAei,x,CxiAeiA≤Bx,xA.
Then,(12)AC1/2K∗x,C1/2K∗xA≤∑i∈Ix,xiAei,Cx,xiAeiA≤Bx,xA.
Therefore,(13)AC1/2K∗x,C1/2K∗xA≤∑i∈IΓix,ΓiCxA≤Bx,xA.
SinceC is a surjective operator, then, from Lemma 1, there exists m>0, such that(14)mK∗x,K∗xA≤C1/2K∗x,C1/2K∗xA.
Then,(15)AmK∗x,K∗xA≤∑i∈IΓix,ΓiCxA≤Bx,xA,which shows that Γii∈I is a Idℋ,C-controlled K-operator frame for EndA∗ℋ.Proposition 1.
EveryC,C′-controlled operator frame for EndA∗ℋ is a C,C′-controlled K-operator frame for EndA∗ℋ.Proof.
LetTii∈I be a C,C′-controlled operator frame for EndA∗ℋ with bounds A and B; then, we have, for all x∈ℋ,(16)Ax,xA≤∑i∈ITiCx,TiC′xA≤Bx,xA.
From Lemma2, we have(17)AK−2K∗x,K∗xA≤∑i∈ITiCx,TiC′xA≤Bx,xA.
Therefore,Tii∈I is a C,C′-controlled K-operator frame for EndA∗ℋ with bounds AK−2 and B.Proposition 2.
LetTii∈I be a C,C′-controlled K-operator frame for EndA∗ℋ. If K is a surjective operator, then Tii∈I is a C,C′-controlled operator frame forEndA∗ℋ.Proof.
LetTii∈I be a C,C′-controlled K-operator frame for EndA∗ℋ with bounds A and B; hence,(18)AK∗x,K∗xA≤∑i∈ITiCx,TiC′xA≤Bx,xA,x∈ℋ.
SinceK is surjective, then, from Lemma 1, there exists m>0 such that(19)mx,xA≤K∗x,K∗xA,x∈ℋ.
Using (18) and (19), we have(20)Amx,xA≤∑i∈ITiCx,TiC′xA≤Bx,xA,x∈ℋ.
Therefore,Tii∈I is a C,C-controlled operator frame for EndA∗ℋ with bounds Am and B.Proposition 3.
LetC,C′∈GL+ℋ and Tii∈I be a K-operator frame for EndA∗ℋ. Assume that C and C′ commutes between them and commutes with the operatorsTi∗TiandK, for eachi∈I. Then,Tii∈Iis aC,C′-controlledK-operator frame forEndA∗ℋ.Proof.
LetTii∈I be a K-operator frame for EndA∗ℋ; then, there exist 0<A,B such that(21)AK∗x,K∗xA≤∑i∈ITix,TixA≤Bx,xA,x∈ℋ.
On the one hand, we have, for allx∈ℋ,(22)∑i∈ITiCx,TiC′xA=∑i∈ITiCC′1/2x,TiCC′1/2xA≤BCC′1/2x,CC′1/2xA.
Then,(23)∑i∈ITiCx,TiC′xA≤BCC′1/22x,xA.
On the other hand, we have, for allx∈ℋ,(24)∑i∈ITiCC′1/2x,TiCC′1/2xA≥AK∗CC′1/2x,K∗CC′1/2xA=ACC′1/2K∗x,CC′1/2K∗xA.
From Lemma1, there exists m>0 such that(25)CC′1/2K∗x,CC′1/2K∗xA≥mK∗x,K∗xA.
From (23) and (24), we have, for all x∈ℋ,(26)AmK∗x,K∗xA≤∑i∈ITiCx,TiC′xA≤BCC′1/22x,xA.
Therefore,Tii∈I is a C,C′-controlled K-operator frame for EndA∗ℋ.
From now, we suppose thatC and C′ commutes between them and commutes with the operators Ti∗Ti for each i∈I. Let Tii∈I be a C,C′-controlled K-operator Bessel sequence for EndA∗ℋ. We define the operator TC,C′ by(27)TC,C′:ℋ⟶l2ℋx⟶TC,C′x=TiCC′1/2xi∈I.
The operatorTC,C′, called the analysis operator, is well defined and bounded. The adjoint operator of TC,C′, called the synthesis operator, is defined by(28)TC,C′∗:l2ℋ⟶ℋaii∈I⟶TC,C′∗aii∈I=∑i∈IC,C′1/2Ti∗ai.
We define the frame operator forTii∈I by(29)SC,C′:ℋ⟶ℋx⟶SC,C′x=TC,C′∗TC,C′x=∑i∈IC′Ti∗TiCx.
It is clear to see thatSC,C is a positive, bounded, and self-adjoint operator.Theorem 2.
LetTii∈I be a C,C′-controlled operator Bessel sequence for EndA∗ℋ. The following statements are equivalent:(1)
Tii∈Iis aC,C′-controlledK-operator frame(2)
There existsA>0such thatAKK∗≤SC,C′(3)
K=SC,C′1/2Q, for someQ∈EndA∗ℋProof.
1⇒2: let Tii∈I be a C,C-controlled K-operator frame for EndA∗ℋ with bounds A and B and frame operator SC,C; then, we have, for all x∈ℋ,(30)AK∗x,K∗xA≤∑i∈ITiCx,TiC′xA≤Bx,xA.
Therefore,(31)AKK∗x,xA≤∑i∈IC′Ti∗TiCx,xA≤Bx,xA,so(32)AKK∗x,xA≤SC,C′x,xA≤Bx,xA.
Hence,(33)AKK∗≤SC,C′.
2⇒3: let A>0 such that(34)AKK∗≤SC,C′.
This gives(35)AKK∗≤SC,C′1/2SC,C′1/2∗.
From Theorem1, we have(36)K=SC,C′1/2Q,for some Q∈EndA∗ℋ.
3⇒1: suppose that(37)K=SC,C′1/2Q,for some Q∈EndA∗ℋ.
From Theorem1, there exists 0<A such that(38)AKK∗≤SC,C′1/2SC,C′1/2∗.
Hence,(39)AKK∗≤SC,C′.
Therefore,Tii∈I is a C,C′−controlled K-operator frame for EndA∗ℋ.Proposition 4.
LetK,Q∈EndA∗ℋ and Tii∈I be a C,C′-controlled K-operator frame for EndA∗ℋ. Suppose that Q commutes with C,C′, andK. Then,TiQi∈Iis aC,C′-controlledQ∗K-operator frame forEndA∗ℋ.Proof.
Suppose thatTii∈I is a C,C′-controlled K-operator frame with bounds A and B; then, we have, for all x∈ℋ,(40)AK∗x,K∗xA≤∑i∈ITiCx,TiC′xA≤Bx,xA.
Hence,(41)AK∗Qx,K∗QxA≤∑i∈ITiCQx,TiC′QxA≤BQx,QxA,so(42)AQ∗K∗x,Q∗K∗xA≤∑i∈ITiQCx,TiQC′xA≤BQ2x,xA.
Therefore,TiQi∈I is a C,C′-controlled Q∗K-operator frame for EndA∗ℋ with bounds A and BQ2.Theorem 3.
LetK∈EndA∗ℋ and Tii∈I be a C,C′-controlled tight K-operator frame for EndA∗ℋ with bound A1. Then, Tii∈I is a C,C′-controlled tight operator frame for EndA∗ℋ with bound A2 if and only if Kr−1=A1/A2K∗.Proof.
LetTii∈I be a C,C′-controlled tight K-operator frame for EndA∗ℋ with bound A1; then, we have, for all x∈ℋ,(43)A1K∗x,K∗xA=∑i∈ITiCx,TiC′xA.
Assume thatTii∈I is a C,C′-controlled tight operator frame for EndA∗ℋ with bound A2; then, we have, for all x∈ℋ,(44)∑i∈ITiCx,TiC′xA=A2x,xA.
Hence,(45)A1K∗x,K∗xA=A2x,xA,so(46)KK∗x,xA=A2A1x,xA.
Then,(47)KK∗=A2A1IdA.
Therefore,(48)Kr−1=A1A2K∗.
For the converse, assume that(49)Kr−1=A1A2K∗.
Then,(50)KK∗=A2A1IdA.
This gives(51)KK∗x,x=A2A1x,xA,x∈ℋ.
OrTii∈I is a C,C′-controlled tight K-operator frame for EndA∗ℋ with bound A1, then we have(52)∑i∈ITiCx,TiC′xA=A2x,xA.
Therefore,Tii∈I is a C,C′-controlled tight operator frame for EndA∗ℋ.Corollary 1.
LetK∈EndA∗ℋandTii∈Ibe a sequence forEndA∗ℋ. Then, those statements are true:IfTii∈Iis aC,C′-controlled tightK-operator frame forEndA∗ℋ, thenTiKn∗i∈Iis aC,C′-controlled tightKn+1-operator frame forEndA∗ℋIfTii∈Iis aC,C′-controlled tight operator frame forEndA∗ℋ, thenTiK∗i∈Iis aC,C′-controlled tightK-operator frame forEndA∗ℋTheorem 4.
LetTii∈I be a C,C′-controlled K-operator frame for EndA∗ℋ with best frame bounds A and B. If Q:ℋ⟶ℋ is an adjointable and invertible operator such thatQ−1commutes withK∗, thenTiQi∈Iis aC,C′-controlled K-operator frame forEndA∗ℋwith best frame boundsMandNsatisfying the inequalities:(53)AQ−1−2≤M≤AQ2andBQ−1−2≤N≤BQ2.Proof.
LetTii∈I be a C,C′-controlled K-operator frame for EndA∗ℋ with best frame bounds A and B; then, for all x∈ℋ, we have(54)∑i∈ITiCQx,TiC′QxA≤BQx,QxA≤BQ2x,xA.
Also, we have(55)AK∗x,K∗xA=AK∗Q−1Qx,K∗Q−1QxA=AQ−1K∗Qx,Q−1K∗QxA≤Q−12∑i∈ITiCQx,TiC′QxA=Q−12∑i∈ITiQCx,TiQC′xA.
Hence,(56)AQ−1−2K∗x,K∗xA≤∑i∈ITiQCx,TiQC′xA≤BQ2x,xA.
Therefore,TiQi∈I is a C,C′-controlled K-operator frame for EndA∗ℋ with bounds AQ−1−2 and BQ2.
Now, letM and N be the best bounds of the C,C′-controlled K-operator frame TiQi∈I. Then,(57)AQ−1−2≤MandN≤BQ2.
SinceTiQi∈I is a C,C′-controlled K-operator frame for EndA∗ℋ with frame bounds M and N, we have, for all x∈ℋ,(58)K∗x,K∗xA=QQ−1K∗x,QQ−1K∗xA≤Q2K∗Q−1x,K∗Q−1xA.
Hence,(59)MQ−2K∗x,K∗xA≤MK∗Q−1x,K∗Q−1xA≤∑i∈JTiQCQ−1x,TiQC′Q−1xA=∑i∈JTiQQ−1Cx,TiQQ−1C′xA=∑i∈ITiCx,TiC′xA≤NQ−12x,xA.
SinceA and B are the best frame bounds of C,C′-controlled K-operator frame Tii∈I, we have(60)MQ−2≤AandB≤NQ−12.
Therefore, inequality (53) follows from (57) and (60).


## 4. Tensor Product of ControlledK-Operator Frame

Theorem 5.
Letℋ and K be two Hilbert C∗-modules over a unital C∗-algebras A and ℬ, respectively. Let C1,C2∈GL+ℋ and C1′,C2′∈GL+K, and letTii∈Ibe aC1,C2-controlledK-operator frame forℋandΓjj∈Jbe aC1′,C2′-controlledL-operator frame forKwith frame operatorsSTandSΓand boundsA1,B1andA2,B2, respectively, whereK∈EndA∗ℋandL∈Endℬ∗K. Then,Ti⊗Γji∈I,j∈Jis aC1⊗C1′,C2⊗C2′-controlledK⊗L-operator frame for HilbertA⊗ℬ-moduleℋ⊗K, with frame operatorST⊗SΓand boundsA1⊗A2andB1⊗B2.Proof.
According to the definitions ofTii∈I and Γjj∈J contained in the previous theorem, we have, for all x∈ℋ and y∈K,(61)A1K∗x,K∗xA≤∑i∈ITiC1x,TiC2xA≤B1x,xAA2L∗y,L∗yℬ≤∑j∈JΓjC1′y,ΓjC2′yℬ≤B2y,yℬ.
Therefore,(62)A1K∗x,K∗xA⊗A2L∗y,L∗yℬ≤∑i∈ITiC1x,TiC2xA⊗∑j∈JΓjC1′y,ΓjC2′yℬ≤B1x,xA⊗B2y,yℬ,which gives(63)A1⊗A2K∗x,K∗xA⊗L∗y,L∗yℬ≤∑i∈I,j∈JTiC1x,TiC2xA⊗ΓjC1′y,ΓjC2′yA)≤≤B1⊗B2x,xA⊗y,yℬ.
Then,(64)A1⊗A2K∗x⊗L∗y,K∗x⊗L∗yA⊗ℬ≤∑i∈I,j∈JTiC1x⊗ΓjC1′y,TiC2x⊗ΓjC2′yA⊗ℬ≤≤B1⊗B2x⊗y,x⊗yA⊗ℬ,so(65)A1⊗A2K∗⊗L∗x⊗y,K∗⊗L∗x⊗yA⊗ℬ≤∑i∈I,j∈JTi⊗ΓjC1x⊗C1′y,Ti⊗ΓjC2x⊗C2′yA⊗ℬ≤B1⊗B2x⊗y,x⊗yA⊗ℬ.
Then,(66)A1⊗A2K⊗L∗x⊗y,K⊗L∗x⊗yA⊗ℬ≤∑i∈I,j∈JTi⊗ΓjC1⊗C1′x⊗y,Ti⊗ΓjC2⊗C2′x⊗yA⊗ℬ≤B1⊗B2x⊗y,x⊗yA⊗ℬ.
Therefore,Ti⊗Γji∈I,j∈J is a C1⊗C1′,C2⊗C2′-controlled K⊗L-operator frame for Hilbert A⊗ℬ-module ℋ⊗K.
Moreover, we have, for allx∈ℋ and y∈K,(67)ST⊗SΓx⊗y=STx⊗SΓy=∑i∈IC2Ti∗TiC1x⊗∑j∈JC2′Γj∗ΓjC1′y=∑i∈I,j∈JC2Ti∗TiC1x⊗C2′Γj∗ΓjC1′y=∑i∈I,j∈JC2⊗C2′Ti∗⊗Γj∗TiC1x⊗ΓjC1′y=∑i∈I,j∈JC2⊗C2′Ti⊗Γj∗Ti⊗ΓjC1x⊗C1′y=∑i∈I,j∈JC2⊗C2′Ti⊗Γj∗Ti⊗ΓjC1⊗C1′x⊗y.
By uniqueness of frame operator, the last equality is equal toST⊗Γx⊗y, which ends the proof.Theorem 6.
Letℋ and K be two Hilbert C∗-modules over a unital C∗-algebras A and ℬ, respectively. Let Q∈EndA∗ℋ and L, C, and C′inEndA⊗ℬ∗ℋ⊗K, whereQis an invertible operator. LetTii∈Ia sequence of adjointable operators inEndA⊗ℬ∗ℋ⊗Kbe aC,C′-controlledL-operator frame forℋ⊗Kwith lower and upper boundsAandB, respectively, and frame operatorST. IfQ⊗Icommutes withL,C, andC′, thenTiQ∗⊗Ii∈Iis aC,C′-controlledL-operator frame forℋ⊗Kwith boundsQ∗−1−2AandQ2Band frame operatorQ⊗ISQ∗⊗I.Proof.
LetQ∈EndA∗ℋ; then, Q⊗I∈EndA⊗ℬ∗ℋ⊗K with inverse Q−1⊗I. Also, it is easy to show that the adjoint of Q⊗I is Q∗⊗I, and we have, for all x⊗y∈ℋ⊗K,(68)Q⊗Ix⊗y≤Q2x⊗y,which show that Q⊗I is bounded. Similarly, for Q∗⊗I, hence, Q⊗I is A⊗ℬ-linear, adjointable with adjoint Q∗⊗I.
So, for allz∈ℋ⊗K, by Lemma 3, we have(69)Q∗−1−1.z≤Q∗⊗Iz≤Q.z.
From the definition ofTii∈I, we have, for all z∈ℋ⊗K,(70)AL∗z,L∗zA⊗ℬ≤∑i∈ITiCz,TiC′zA⊗ℬ≤Bz,zA⊗ℬ.
Then,(71)AL∗Q∗⊗Iz,L∗Q∗⊗IzA⊗ℬ≤∑i∈ITiCQ∗⊗Iz,TiC′Q∗⊗IzA⊗ℬ≤BQ∗⊗Iz,Q∗⊗IzA⊗ℬ.
On the one hand, we have(72)BQ∗⊗Iz,Q∗⊗IzA⊗ℬ≤Q2Bz,zA⊗ℬ.
On the other hand, we have(73)AL∗Q∗⊗Iz,L∗Q∗⊗IzA⊗ℬ=AQ∗⊗IL∗z,Q∗⊗IL∗zA⊗ℬ≥Q∗−1−2AL∗z,L∗zA⊗ℬ.
Finally, we obtain(74)Q∗−1−2AL∗z,L∗zA⊗ℬ≤∑i∈ITiQ∗⊗ICz,TiQ∗⊗IC′zA⊗ℬ≤Q2Bz,zA⊗ℬ.
Concerning the frame operator, we have(75)Q⊗ISQ∗⊗I=Q⊗I∑i∈IC′Ti∗TiCQ∗⊗I=∑i∈IQ⊗IC′Ti∗TiCQ∗⊗I=∑i∈IC′Q⊗ITi∗TiQ∗⊗IC=∑i∈IC′TiQ∗⊗Ii∗TiQ∗⊗IC,which allows us to find the requested result.Theorem 7.
Letℋ and K be two Hilbert C∗-modules over a unital C∗-algebras A and ℬ, respectively. Let Q∈EndA∗ℋ and L, C, and C′be inEndA⊗ℬ∗ℋ⊗K, whereQis an invertible operator. LetTii∈Ia sequence of adjointable operators inEndA⊗ℬ∗ℋ⊗Kbe aC,C′-controlledL-operator frame forℋ⊗Kwith lower and upper boundsAandB, respectively, and frame operatorST. IfI⊗Qcommutes withL,C, andC′, thenTiI⊗Q∗i∈Iis aC,C′-controlledL-operator frame forℋ⊗Kwith boundsQ∗−1−2AandQ2Band frame operatorQ⊗ISQ∗⊗I.Proof.
It is similar to the proof of the previous theorem.Theorem 8.
Letℋ,A,.,.A and ℋ,ℬ,.,.ℬ be two Hilbert C∗-modules and let φ:A⟶ℬ between C∗-algebras be a ∗-homomorphism andθbe a map onℋsuch thatθx,θyℬ=φx,yA, for allx,y∈ℋ. Also, suppose thatTii∈Iis aC,C′-controlledL-operator frame forEndA∗ℋwith frame operatorSA. Ifθis surjective and commutes withL∗,Ti,C, andC′, thenTii∈Iis aC,C′-controlledL-operator frame forEndℬ∗ℋwith frame operatorSℬandSℬθx,θyℬ=φSAx,yA.Proof.
Letz∈ℋ; since θ is surjective, then there exist z∈ℋ such that θx=z. By the definition of C,C′-controlled L-operator frame Tii∈I for ℋ, there exist 0<A,B such that(76)AL∗x,L∗xA≤∑i∈ITiCx,TiC′xA≤Bx,xA.
By Lemma4, we have(77)φAL∗x,L∗xA≤φ∑i∈ITiCx,TiC′xA≤φBx,xA⇒AφL∗x,L∗xA≤∑i∈IφTiCx,TiC′xA≤Bφx,xA⇒AθL∗x,θL∗xℬ≤∑i∈IθTiCx,θTiC′xA≤Bθx,θxℬ⇒AL∗θx,L∗θxℬ≤∑i∈ITiCθx,TiC′θxA≤Bθx,θxℬ⇒AL∗z,L∗zℬ≤∑i∈ITiCz,TiC′zA≤Bz,zℬ,which shows that Tii∈I is a C,C′-controlled L-operator frame for Endℬ∗ℋ.
Moreover, letSA and Sℬ be the frame operators for Tii∈I and θTii∈I, respectively; then, we have(78)φSAx,yA=φ∑i∈IC′Ti∗TiCx,yA=∑i∈IφC′Ti∗TiCx,yA=∑i∈IφTiCx,TiC′yA=∑i∈IθTiCx,θTiC′yA)=∑i∈ITiCθx,TiC′θyA)=∑i∈IC′Ti∗TiCθx,θyA=Sℬθx,θyℬ,which ends the proof.



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*Source: 7296689-2022-03-14.xml*