# Controlled∗-Operator Frames on Hilbert C∗-Modules

Authors: Hatim Labrigui; Abdeslam Touri; Mohamed Rossafi; Samir Kabbaj
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we study the concept of controlled∗-operator frames for the space of all adjointable operators on a Hilbert C∗-module H. Also, we discuss characterizations of controlled ∗-operator frames and we give some properties. Some illustrative examples are provided to advocate the usability of our results.


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## Body

## 1. Introduction and Preliminaries

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 Daubechies et al., 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.Controlled frames in Hilbert spaces have been introduced by Balazs [4] to improve the numerical efficiency of iterative algorithms for inverting the frame operator.Controlled frames in HilbertC∗-modules were introduced by Rashidi and Rahimi [5], and the authors showed that they share many useful properties with their corresponding notions in a Hilbert space. ∗-operator frames for EndA∗ℋ has been studied by Rossafi and Kabbaj [6]. For more details, see [7–9].In this paper, we introduce the notion of controlled∗-operator frames for EndA∗ℋ, where ℋ is a Hilbert C∗-modules.LetI be a countable index set. In this section, we briefly recall the definitions and basic properties of C∗-algebra, Hilbert C∗-modules, frame, and operator frame in Hilbert C∗-modules. For information about frames in Hilbert spaces, we refer to [10]. Our references for C∗-algebras are [11, 12].For aC∗-algebra A, an element a∈A is positive (a≥0) if a=a∗ and spa⊂R+. A+ denotes the set of positive elements of A.Definition 1.
(see [13]). Let A 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 sesquilinear, positive definite, and respects the module action. In the 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 x=x,xA1/2. If ℋ is complete with ., it is called a Hilbert A-module or a Hilbert C∗-module over A.
For everya in C∗-algebra A, we have a=a∗a1/2 and the A-valued norm on ℋ is defined by x=x,xA1/2 for x∈ℋ.Example 1.
(see [14]). If ℋkk∈N is a countable set of Hilbert A-modules, then one can define their direct sum ⊕k∈ℕℋk. On the A-module ⊕k∈ℕℋk of all sequences x=xkk∈ℕ:xk∈ℋk, such that the series ∑k∈ℕxk,xkA is norm-convergent in the C∗-algebra A, we define the inner product by(1)x,y≔∑k∈ℕxk,ykA,for x,y∈⊕k∈ℕℋk.
Hence,⊕k∈ℕℋk is a Hilbert A-module.
The direct sum of a countable number of copies of a HilbertC∗-module ℋ is denoted by l2ℋ.
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 also 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.
LetGL+ℋ be the set for all positive bounded linear invertible operators on ℋ with bounded inverse.
The following lemmas will be used to prove our main results.Lemma 1 (see [15]).
Ifφ:A⟶ℬ is a ∗-homomorphism between C∗-algebras, then φ is increasing, that is, if a≤b, then φa≤φb.Lemma 2 (see [15]).
Letℋ and K be two Hilbert A-modules and T∈EndA∗ℋ,K.(i)
IfT is injective and T has closed range, then the adjointable map T∗T is invertible and(2)T∗T−1−1Iℋ≤T∗T≤T2Iℋ.(ii)
IfT is surjective, then the adjointable map TT∗ is invertible and(3)TT∗−1−1IK≤TT∗≤T2IK.Lemma 3 (see [16]).
Letℋ be a Hilbert A-modules. If T∈EndA∗ℋ, then(4)Tx,TxA≤T2x,xA,x∈ℋ.Lemma 4 (see [17]).
Letℋ and K be two Hilbert A-modules and T∈End∗ℋ,K. Then, the following statements are equivalent:(i)
T is surjective(ii)
T∗ is bounded below with respect to norm, i.e., there is m>0 such that mx≤T∗x for all x∈K(iii)
T∗ is bounded below with respect to the inner product, i.e., there is m′>0 such that m′x,xA≤T∗x,T∗xA for all x∈K


## 2. Controlled∗-Operator Frames for EndA∗ℋ

We begin this section with the following definition.Definition 2.
(see [6]). A family of adjointable operators Tii∈I on a Hilbert A-module ℋ over a unital C∗-algebra A is said to be an operator frames for EndA∗ℋ, if there exist two positive constants A,B>0 such that(5)Ax,xA≤∑i∈ITix,Tix≤Bx,xA,x∈ℋ.
The numbersA and B are called lower and upper bound of the operator frames, respectively. If A=B=λ, the operator frame is λ-tight.
IfA=B=1, it is called a normalized tight operator frames or a Parseval operator frames.
If only upper inequality of (5) holds, then Tii∈I is called an operator Bessel sequence for EndA∗ℋ.
If the sum in the middle of (5) is convergent in norm, the operator frame is called standard.Definition 3.
LetC,C′∈GL+ℋ, and a family 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 ∗- operator frames for EndA∗ℋ if there exists two strictly nonzero elements A and B in A such that(6)Ax,xAA∗≤∑i∈ITiCx,TiC′x≤Bx,xAB∗,x∈ℋ.
The elementsA and B are called lower and upper bound of the C,C′-controlled ∗-operator frames, respectively.
IfA=B=λ, the C,C′-controlled ∗-operator frame is called λ-tight.
IfA=B=1A, it is called a normalized tight C,C′-controlled ∗-operator frames or a Parseval C,C′-controlled ∗-operator frames.
If only upper inequality of (6) holds, then Tii∈i is called an C,C′-controlled ∗-operator Bessel sequence for EndA∗ℋ.Example 2.
LetA=l∞ be the unitary C∗-algebra of all bounded complex-valued sequences and let ℋ=C0, the set of all sequences converging to zero equipped with the A=l∞-inner product:(7)x,yA=xii∈ℕ,yii∈ℕA=xiy¯ii∈ℕ,for allx=xii∈ℕandy=yii∈ℕ∈ℋ.
It is clear to see thatℋ is a Hilbert C∗-module over A=l∞.
Letj∈ℕ and aii∈ℕ=1+1/ii∈ℕ, and we define Tj∈EndA∗ℋ by(8)Tjxii∈ℕ=δijajxji∈ℕ,xii∈ℕ∈ℋ.
Letα∈ℝ∗, and we define two operators C and C′ on ℋ by(9)C:ℋ⟶ℋ,x⟶Cx=αx,C′:ℋ⟶ℋ,x⟶C′x=1αx.
We have(10)∑j∈ℕTjCx,TjC′x=δijajαxji∈ℕ,δijaj1αxji∈ℕA=1+1i2xix¯ii∈ℕ=1+1ii∈ℕx,xA1+1ii∈ℕ.
Therefore,Tjj∈ℕ is a C,C′-controlled tight ∗-operator frame for EndA∗ℋ.Proposition 1.
EveryC,C′-controlled operator frames for EndA∗ℋ is a C,C′-controlled ∗-operator frames.Proof.
LetTii∈I be a C,C′-controlled ∗-operator frames for EndA∗ℋ.
Then, there exist two positives constantsA,B>0 such that(11)Ax,xA≤∑i∈ITiCx,TiC′xA≤Bx,xA,x∈ℋ.
Hence,(12)A1Ax,xAA1A∗≤∑i∈ITiCx,TiC′xA≤B1Ax,xAB1A∗,x∈ℋ.
Therefore,Tii∈I is a C,C′-controlled ∗-operator frames for EndA∗ℋ with bounds A1A and B1A.
LetTii∈I be a C,C′-controlled ∗-operator frame for EndA∗ℋ.
The bounded linear operatorTC,C′:ℋ⟶l2ℋ is given by(13)TC,C′x=TiC′C1/2xi∈I,x∈ℋ,which is called the analysis operator for the C,C′-controlled ∗-operator frame Tii∈I.
The adjoint operatorTCC′∗:l2ℋ⟶ℋ is given by(14)TC,C′∗yii∈I=∑i∈ICC′1/2Ti∗yi,yii∈I∈l2ℋ,which is called the synthesis operator for the C,C′-controlled ∗-operator frames Tii∈I.
WhenC and C′ commute with each other and commute with the operator Ti∗Ti for each i∈I, we define the C,C′-controlled frame operator:SC,C′:ℋ⟶ℋ by SC,C′x=TC,C′TC,C′∗x=∑i∈IC′Ti∗TiCx.
From now, we assume thatC and C′ commute with each other and commute with the operator Ti∗Ti for each i∈I.Proposition 2.
TheC,C′-controlled frame operator SC,C′ is bounded, positive, self-adjoint, and invertible.Proof.
LetTii∈I be a C,C′-controlled ∗-operator frames; then,(15)Ax,xAA∗≤∑i∈ITiCx,TiC′xA≤Bx,xAB∗,and hence,(16)∑i∈ITiCx,TiC′xA=∑i∈IC′Ti∗TiCx,xA=SCC′x,xA.
Then,(17)Ax,xAA∗≤SCC′x,xA≤Bx,xAB∗.
It is clear thatSCC′ is positive and bounded operator.
We have(18)SC,C′x,yA=∑i∈IC′Ti∗TiCx,yA=∑i∈IC′Ti∗TiCx,yA=∑i∈Ix,CTi∗TiC′yA=x,∑i∈ICTi∗TiC′yA=x,SC′,CyA.
Therefore,SC,C′∗=SC′,C. Since C and C′ commute with each other and commute with Ti∗Ti, we have SC,C′ self-adjoint.
From the definition of controlled∗-operator frame, we have(19)Ax,xAA∗≤SC,C′x,xA≤Bx,xAB∗.
So,(20)A.Idℋ.A∗≤SC,C′≤B.Idℋ.B∗,where Idℋ is the identity operator in ℋ. Thus, SC,C′ is invertible.


## 3. Some Characterizations of Controlled∗-Operator Frames

Theorem 1.
LetTii∈I⊂EndA∗ℋ such that ∑i∈ITiCx,TiC′xA converge in norm A. Then, Tii∈I is a C,C′-controlled ∗-operator frames if and only if(21)A−1−2x,xA≤∑i∈ITiCx,TiC′xA≤B2x,xA,for every x∈ℋ and strictly nonzero elements A,B in A.Proof.
Suppose thatTii∈I is a C,C′-controlled ∗-operator frames; then, we have(22)x,xA≤A−1SC,C′x,xAA∗−1,x∈ℋ,SC,C′x,xA≤Bx,xAB∗,x∈ℋ.
Hence,(23)A−1−2x,xA≤∑i∈ITiCx,TiC′xA≤B2x,xA.
Conversely, assume that (21) holds. From (13), the C,C′-controlled frame operator SC,C′ is positive, self-adjoint, and invertible. Then, we have, for all x∈ℋ,(24)SC,C′1/2x,SC,C′1/2xA=SC,C′x,xA=∑i∈ITiCx,TiC′xA.
Using (21) and (24), we obtain(25)A−1−2⋅x≤SC,C′1/2x≤B2⋅x.
Using (25) and Lemma 4, we conclude that Tii∈I is a C,C′-controlled ∗-operator frames EndA∗ℋ.
The following theorem shows that any∗-operator frame is a C2-controlled ∗-operator frames for ℋ and vice versa.Theorem 2.
LetC∈GL+ℋ. The family Tii∈I∈EndA∗ℋ is a ∗-operator frame for EndA∗ℋ if and only if Tii∈I is a C2-controlled ∗-operator frames.Proof.
LetTii∈I be a C2-controlled ∗-operator frames with bounds A and B. Then,(26)∑i∈ITiCx,TiCxA≤Bx,xAB∗,x∈ℋ.
On the one hand, for allx∈ℋ, we have(27)Ax,xAA∗=ACC−1x,CC−1xAA∗≤AC2C−1x,C−1xAA∗≤C2∑i∈ITiCC−1x,TiCC−1xA=C2∑i∈ITix,TixA.
Then,(28)AC−1x,xAA∗C−1≤∑i∈ITix,TixA.
On the other hand, for anyx∈ℋ, we have(29)∑i∈ITix,TixA=∑i∈ITiCC−1x,TiCC−1xA≤BC−1x,C−1xAB∗≤BC−12x,xAB∗.
Then,(30)AC−1x,xAA∗C−1≤∑i∈ITix,TixA≤BC−1x,xAB∗C−1.
Therefore,Tii∈I is a ∗-operator frames with bounds AC−1 and BC−1.
For the converse, suppose thatTii∈I is a ∗-operator frames with bounds M and N.
On the one hand, we have, for anyx∈ℋ,(31)Mx,xAM∗≤∑i∈ITix,TixA≤Nx,xAN∗.
Thus, for allx∈ℋ, we have(32)∑i∈ITiCx,TiCxA≤NC2x,xAN∗=NCx,xAN∗C.
On the other hand, we have(33)Mx,xAM∗=MC−1Cx,C−1CxAM∗≤C−12∑i∈ITiCx,TiCxA.
Therefore,(34)MC−1−1x,xAM∗C−1−1≤∑i∈ITiCx,TiCxA≤NCx,xAN∗C.
This gives thatTii∈I is a C2-controlled ∗-operator frame with bounds MC−1−1 and N∗C.Proposition 3.
LetTii∈I be a ∗-operator frames for EndA∗ℋ with frame operator S and let C,C′∈GL+ℋ. Suppose that C,C′ commute with each other and commute with TiTi∗ for all i∈I; then, Tii∈I is a C,C′-controlled ∗-operator frame for EndA∗ℋ.Proof.
LetTii∈I be an ∗-operator frame with bounds A and B. Then, by (5), we have(35)x,xA≤A−1Sx,xAA∗−1,Sx,xA≤Bx,xAB∗.
Hence,(36)A−1−2x,xA≤∑i∈ITix,TixA≤B2x,xA.
We have(37)∑i∈ITiCx,TiC′xA=SC,C′x,xA,(38)∑i∈ITiCx,TiC′xA=C⋅C′⋅Sx,xA.
Using (36) and (38), we have(39)A−1−2C⋅C′x,xA≤∑i∈ITiCx,TiC′xA≤B2C⋅C′x,xA.
Therefore, from Theorem1, we conclude that Tii∈I is a C,C′-controlled ∗-operator frame with bounds A−1−1C1/2⋅C′1/2 and BC1/2⋅C′1/2.Theorem 3.
LetC,C′∈GL+ℋ and Tii∈I⊂EndA∗ℋ. Suppose that C,C′ commute with each other and commute with TiTi∗ for all i∈I. The family Tii∈I is a C,C′-controlled ∗-operator Bessel sequence for EndA∗ℋ with bound B if and only if the operator TCC′:l2ℋ⟶ℋ given by(40)TCC′yii∈I=∑i∈ICC′1/2Ti∗yi,∀yii∈I∈l2ℋ,is well defined, bounded, and TCC′≤B.Proof.
Assume thatTii∈I is a C,C′-controlled ∗-operator Bessel sequence for EndA∗ℋ with bound B. As a result of (36),(41)∑i∈ITiCx,TiC′xA≤B2x,xA.
We have(42)TC,C′yii∈I2=supx∈ℋ,x=1∑i∈ICC′1/2Ti∗yi,xA2=supx∈ℋ,x=1∑i∈ICC′1/2Ti∗yi,xA2=supx∈ℋ,x=1∑i∈Iyi,TCC′1/2xA2≤supx∈ℋ,x=1∑i∈Iyi,yiA∑i∈ITiCC′1/2x,TiCC′1/2xA=supx∈ℋ,x=1∑i∈Iyi,yiA∑i∈ITiCx,TiC′xA≤supx∈ℋ,x=1∑i∈Iyi,yiAB2x2=B2yii∈I2.
Then, the sum∑i∈ICC′1/2Ti∗yi is convergent, and we have(43)TC,C′yii∈I2≤B2yii∈I2.
Hence,(44)TC,C′2≤B2.
Thus, the operatorTC,C′ is well defined, bounded, and(45)TC,C′≤B.
For the converse, suppose that the operatorTC,C′ is well defined, bounded, and TC,C′≤B. For all x∈ℋ, we have(46)∑i∈ITiCx,TiC′xA=∑i∈IC′Ti∗TiCx,xA=∑i∈ICC′1/2Ti∗TiCC′1/2x,xA=TC,C′yii∈I,xA≤TC,C′yii∈Ix≤TC,C′∑i∈ITiCC′1/2x21/2x=TC,C′∑i∈ITiCx,TiC′xA1/2x,where yi=TiCC′1/2x.
Therefore,(47)∑i∈ITiCx,TiC′xA≤TC,C′2x2.
Hence,(48)∑i∈ITiCx,TiC′xA≤B2x2,asTC,C′≤B.
This give thatTii∈I is a C,C′-controlled ∗-operator Bessel sequence for EndA∗ℋ.Theorem 4.
LetTii∈I⊂EndA∗ℋ be a C,C′-controlled ∗-operator frames for EndA∗ℋ with bounds A and B, with operator frame SC,C′. Let θ∈EndA∗ℋ be injective and has a closed range. Suppose that θ commute with C and C′. Then, Tiθi∈I is a C,C′-controlled ∗-operator frames for EndA∗ℋ with frame operator θ∗SC,C′θ with bounds θ∗θ−1−1/2A and θB.Proof.
LetTii∈I⊂EndA∗ℋ be a C,C′-controlled ∗-operator frames for EndA∗ℋ with bounds A and B; then,(49)Aθx,θxAA∗≤∑i∈ITiCθx,TiC′θxA≤Bθx,θxAB∗.
From Lemma2, we have(50)θ∗θ−1−1x,xA≤θx,θxA,x∈ℋ.
Hence,(51)θ∗θ−1−1/2Ax,xAθ∗θ−1−1/2A∗≤Aθx,θxAA∗.
Since(52)θx,θxA≤θ2x,xA,we have(53)Bθx,θxAB∗≤θBx,xAθB∗,x∈ℋ.
Using (49), (51), and (53), we have(54)θ∗θ−1−1/2Ax,xAθ∗θ−1−1/2A∗≤∑i∈ITiCθx,TiC′θxA≤θBx,xAθB∗,x∈ℋ.
Therefore,Tiθi∈I is a C,C′-controlled ∗-operator frame for EndA∗ℋ. Moreover, for every x∈ℋ, we have(55)θ∗SC,C′θ=θ∗∑i∈IC′Ti∗TiCθx=∑i∈Iθ∗C′Ti∗TiCθx=∑i∈IC′Tiθ∗TiθCx.
This completes the proof.Corollary 1.
LetTii∈I⊂EndA∗ℋ be a C,C′-controlled ∗-operator frame for EndA∗ℋ, with frame operator SC,C′. Then, TiSC,C′−1i∈I is a C,C′-controlled ∗-operator frame for EndA∗ℋ.Proof.
The proof is a result of (53) for θ=S−1.Theorem 5.
LetTii∈I⊂EndA∗ℋ be a C,C′-controlled ∗-operator frame for EndA∗ℋ with bounds A and B. Let θ∈EndA∗ℋ be surjective. Then, θTii∈I is a C,C′-controlled ∗-operator frame for EndA∗ℋ with bounds Aθθ∗−1−1/2 and Bθ.Proof.
From the definition ofC,C′-controlled ∗-operator frame, we have(56)Ax,xAA∗≤∑i∈ITiCx,TiC′xA≤Bx,xAB∗,x∈ℋ.
Using Lemma2, we have(57)θθ∗−1−1TiCx,TiC′xA≤θTiCx,θTiC′xA≤θ2TiCx,TiC′xA.
From (56) and (57), we have(58)Aθθ∗−1−1/2x,xAAθθ∗−1−1/2∗≤∑i∈IθTiCx,θTiC′x≤Bθx,xABθ∗,x∈ℋ.
Hence,θTii∈I is a C,C′-controlled ∗-operator frame for EndA∗ℋ.
Under those conditions, a controlled∗- operator frame for EndA∗ℋ with ℋ a C∗-module over a unital C∗-algebras A is also a controlled ∗-operator frame for EndA∗ℋ with ℋ a C∗-module over a unital C∗-algebras ℬ. The following theorem answers these questions.Theorem 6.
Letℋ,A,.,.A and ℋ,ℬ,.,.ℬ be two hilbert C∗-modules and let φ: A⟶ℬ be a ∗-homomorphism and θ be a map on ℋ such that θx,θyℬ=φx,yA for all x,y∈ℋ. Suppose Tii∈I⊂EndA∗ℋ is a C,C′-controlled ∗-operator frame for ℋ,A,.,.A with frame operator SA and lower and upper bounds A and B, respectively. If θ is surjective such that θTi=Tiθ for each i∈I and θC=Cθ and θC′=C′θ, then Tii∈I is a C,C′-controlled ∗-operator frame for ℋ,ℬ,.,.ℬ with frame operator Sℬ and lower and upper bounds φA and φB, respectively, and Sℬθx,θyℬ=φSAx,yA.Proof.
Sinceθ is surjective, then, for every y∈ℋ, there exists x∈ℋ such that θx=y. Using the definition of C,C′-controlled ∗-operator frame, we have(59)Ax,xAA∗≤∑i∈ITiCx,TiC′x≤Bx,xAB∗,x∈ℋ.
By Lemma1, we have(60)φAx,xAA∗≤φ∑i∈ITiCx,TiC′xA≤φBx,xAB∗,x∈ℋ.
From the definition of∗-homomorphism, we have(61)φAφx,xAφA∗≤φ∑i∈ITiCx,TiC′xA≤φBφx,xAφB∗,x∈ℋ.
Using the relation betweenθ and φ, we obtain(62)φAθx,θxℬφA∗≤∑i∈IθTiCx,θTiC′xℬ≤φBθx,θxℬφB∗,x∈ℋ.
SinceθTi=Tiθ, θC=Cθ, and θC′=C′θ, we have(63)φAθx,θxℬφA∗≤∑i∈ITiCθx,TiC′θxℬ≤ϕBθx,θxℬφB∗,x∈ℋ.
Therefore,(64)φAy,yℬφA∗≤∑i∈ITiCy,TiC′yℬ≤φBy,yℬφB∗,y∈ℋ.
This implies thatTii∈I is a C,C′-controlled ∗-operator frame for ℋ,ℬ,.,.ℬ with bounds φA and φB. Moreover, we have(65)φSAx,yA=φ∑i∈ITiCx,TiC′yA=∑i∈IφTiCx,TiC′yA=∑i∈IθTiCx,θTiC′yℬ=∑i∈ITiCθx,TiC′θyℬ=∑i∈IC′Ti∗TiCθx,θyℬ=Sℬθx,θyA,which completes the proof [18].



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