# Some Generalizations of Frames in Hilbert Modules

Authors: Mohamed Rossafi; Samir Kabbaj
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

Frames play significant role in various areas of science and engineering. In this paper, we introduce the concept of frames for the set of all adjointable operators fromℋ to K and their generalizations. Moreover, we obtain some new results for generalized frames in Hilbert modules.


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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].Traditionally, frames have been used in signal processing, image processing, data compression, and sampling theory. A discreet frame is a countable family of elements in a separable Hilbert space which allows for a stable, not necessarily unique, decomposition of an arbitrary element into an expansion of the frame elements.Many generalizations of the concept of frame have been defined in Hilbert spaces and HilbertC∗-modules [4–9].This paper generalizes the papers’ G-frames as special frames [10], Frames and Operator Frames for Bℋ [11], and Generalized Frames for Bℋ,K [12], in framework of the Hilbert C∗-modules.LetI be a finite or countable index subset of ℕ. In this section, we briefly recall the definitions and basic properties of C∗-algebra, Hilbert A-modules, frames in Hilbert A-modules, and their generalizations. For information about frames in Hilbert spaces, we refer to [13]. Our references for C∗-algebras are [14, 15]. For a C∗-algebra A, if a∈A is positive, we write a≥0, and A+ denotes the closed cone of positive elements in A.Definition 1.
(see [14]). If A is a Banach algebra, an involution is a map a⟼a∗ of A into itself such that, for all a and b in A and all scalar α, the following conditions hold:(1)
a∗∗=a(2)
ab∗=b∗a∗(3)
αa+b∗=α¯a∗+b∗Definition 2.
(see [14]). A C∗-algebra A is a Banach algebra with involution such that(1)a∗a=a2,for every a in A.Examples 1.
(1)
Bℋ, the algebra of bounded operators on a Hilbert space ℋ, is a C∗-algebra, where, for each operator A, A∗ is the adjoint of A(2)
CX, the algebra of continuous functions on a compact space X, is an abelian C∗-algebra, where f∗x:=fx¯(3)
C0X, the algebra of continuous functions on a locally compact space X that vanishes at infinity, is an abelian C∗-algebra, where f∗x:=fx¯Definition 3.
(see [16]). 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 .,.:ℋ×ℋ⟶A, such that it is sesquilinear and positive definite and respects the module action. In other words,(1)
x,x≥0, for all x∈ℋ and x,x=0, if and only if x=0(2)
ax+y,z=ax,z+y,z, for all a∈A and x,y,z∈ℋ(3)
x,y=y,x∗, for all x,y∈ℋ
Forx∈ℋ, we define x=x,x1/2. If ℋ is complete with ., it is called a Hilbert A-module or a Hilbert C∗-module over A. For every a in C∗-algebra A, we have a=a∗a1/2, and the A-valued norm on ℋ is defined by x=x,x1/2 for x∈ℋ.Examples 2.
LetX be a locally compact Hausdorff space and ℋ a Hilbert space, and the Banach space C0X,ℋ of all continuous ℋ-valued functions vanishing at infinity is a Hilbert C∗-module over the C∗-algebra C0X with inner product f,gx:=fx,gx and module operation ϕfx=ϕxfx, for all ϕ∈C0X and f∈C0X,ℋ. 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(2)x,y≔∑k∈ℕxk,ykA,for x,y∈⊕k∈ℕℋk. Then, ⊕k∈ℕℋk is a Hilbert A-module. The direct sum of a countable number of copies of a Hilbert C∗-module ℋ is denoted by l2ℋ.
Letℋ and K be two Hilbert A-modules, and 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.
EndA∗ℋ,K is a Hilbert EndA∗K-module with the inner product T,S=TS∗, ∀T,S∈Bℋ,K.Lemma 1.
(see [17]). LetℋandKbe two HilbertA-modules andT∈EndA∗ℋ,K.(i)
IfTis injective andThas closed range, then the adjointable mapT∗Tis invertible and(3)T∗T−1−1Iℋ≤T∗T≤T2Iℋ.(ii)
IfTis surjective, then the adjointable mapTT∗is invertible and(4)TT∗−1−1IK≤TT∗≤T2IK.Definition 4.
(see [18]). We call a sequence Λi∈EndA∗ℋ,Vi:i∈I a g-frame in Hilbert A-module ℋ with respect to Vi:i∈I if there exist two positive constants C, D, such that, for all x∈ℋ,(5)Cx,xA≤∑i∈IΛix,ΛixA≤Dx,xA,where the numbers C and D are called lower and upper bounds of the g-frame, respectively. If C=D=λ, the g-frame is λ-tight. If C=D=1, it is called a g-Parseval frame. If the sum in the middle of (5) is convergent in norm, the g-frame is called standard.Definition 5.
(see [17]). Let ℋ be a Hilbert A-module over a unital C∗-algebra. A family xii∈I of elements of ℋ is a ∗-frame for ℋ, if there exist strictly nonzero elements A, B in A, such that, for all x∈ℋ,(6)Ax,xAA∗≤∑i∈Ix,xiAxi,xA≤Bx,xAB∗,where the elements A and B are called lower and upper bounds of the ∗-frame, respectively. If A=B=λ1, the ∗-frame is λ1-tight. If A=B=1, it is called a normalized tight ∗-frame or a Parseval ∗-frame. If the sum in the middle of (6) is convergent in norm, the ∗-frame is called standard.Definition 6.
(see [19]). We call a sequence Λi∈EndA∗ℋ,Vi:i∈I a ∗-g-frame in Hilbert A-module ℋ over a unital C∗-algebra with respect to Vi:i∈I if there exist strictly nonzero elements A, B in A, such that, for all x∈ℋ,(7)Ax,xAA∗≤∑i∈IΛix,ΛixA≤Bx,xAB∗,where the elements A and B are called lower and upper bounds of the ∗-g-frame, respectively. If A=B=λ1, the ∗-g-frame is λ1-tight. If A=B=1, it is called a ∗-g-Parseval frame. If the sum in the middle of (7) is convergent in norm, the ∗-g-frame is called standard.Definition 7.
(see [20]). Let K∈EndA∗ℋ and Λi∈EndA∗ℋ,Vi, for all i∈I; then, Λii∈I is said to be a K-g-frame for ℋ with respect to Vii∈I if there exist two constants C,D>0 such that(8)CK∗x,K∗xA≤∑i∈IΛix,ΛixA≤Dx,xA,∀x∈ℋ.
The numbersC and D are called K-g-frame bounds. Particularly, if(9)CK∗x,K∗x=∑i∈IΛix,Λix,∀x∈ℋ.
TheK-g-frame is C-tight.Definition 8.
(see [21]). Let K∈EndA∗ℋ. A family xii∈I of elements of ℋ is a ∗-K-frame for ℋ if there exist strictly nonzero elements A and B in A, such that, for all x∈ℋ,(10)AK∗x,K∗xAA∗≤∑i∈Ix,xiAxi,xA≤Bx,xAB∗,where the elements A and B are called lower and upper bound of the ∗-K-frame, respectively.Definition 9.
(see [22]). Let K∈EndA∗ℋ. We call a sequence Λi∈EndA∗ℋ,ℋi:i∈I a ∗-K-g-frame in Hilbert A-module ℋ with respect to ℋi:i∈I if there exist strictly nonzero elements A and B in A such that(11)AK∗x,K∗xAA∗≤∑i∈IΛix,ΛixA≤Bx,xAB∗,∀x∈ℋ.
The numbersA and B are called lower and upper bounds of the ∗-K-g-frame, respectively. If(12)AK∗x,K∗xA∗=∑i∈IΛix,Λix,∀x∈ℋ,The ∗-K-g-frame is A-tight.


## 2. Some Results for Generalized Frames in HilbertA-Modules

We begin this section with the following hTeorem.Theorem 1.
Letℋbe a HilbertA-module over a commutativeC∗-algebra. A familyxii∈Iof elements ofℋis a∗-frame forℋif and only if there exist strictly positive elementsAandBinA, such that, for allx∈ℋ,(13)Ax,xA≤∑i∈Ix,xiAxi,xA≤Bx,xA.Proof.
Letxii∈I be a ∗-frame for ℋ; then, there exist strictly nonzero elements a and b in A, such that, for all x∈ℋ,(14)ax,xAa∗≤∑i∈Ix,xiAxi,xA≤bx,xAb∗.
By commutativity, we have(15)aa∗x,xA≤∑i∈Ix,xiAxi,xA≤bb∗x,xA.
We poseA=aa∗ and B=bb∗; then, A and B are strictly positive elements in A. Hence,(16)Ax,xA≤∑i∈Ix,xiAxi,xA≤Bx,xA.
Conversely, letA and B be strictly positive elements in A, such that, for all x∈ℋ,(17)Ax,xA≤∑i∈Ix,xiAxi,xA≤Bx,xA,where A and B are strictly positive elements in A; then, there exist a and b in A such that A=aa∗ and B=bb∗.
So,(18)aa∗x,xA≤∑i∈Ix,xiAxi,xA≤bb∗x,xA.
By commutativity, we have(19)ax,xAa∗≤∑i∈Ix,xiAxi,xA≤bx,xAb∗.
Therefore,xii∈I is an ∗-frame for ℋ.Corollary 1.
A sequenceΛi∈EndA∗ℋ,Vi:i∈Iis a∗-g-frame in HilbertA-moduleℋover a commutativeC∗-algebra with respect toVi:i∈Iif and only if there exist strictly positive elementsA,BinA, such that for allx∈ℋ,(20)Ax,xA≤∑i∈IΛix,ΛixA≤Bx,xA.Corollary 2.
LetK∈EndA∗ℋ. A familyxii∈Iof elements ofℋis a∗-K-frame forℋif and only if there exist strictly positive elementsAandBinA, such that, for allx∈ℋ,(21)AK∗x,K∗xA≤∑i∈Ix,xiAxi,xA≤Bx,xA.Corollary 3.
LetK∈EndA∗ℋ. A sequenceΛi∈EndA∗ℋ,ℋi:i∈Iis a∗-K-g-frame in HilbertA-moduleℋover a commutativeC∗-algebra, with respect toℋi:i∈Iif and only if there exist strictly positive elementsAandBinA, such that, for allx∈ℋ,(22)AK∗x,K∗xA≤∑i∈IΛix,ΛixA≤Bx,xA.
For a sequence of HilbertA-module Kii∈I, the space ⊕i∈IKi is a Hilbert A-module with the inner product(23)xii∈I,yii∈I=∑i∈Ixi,yiA.Proposition 1.
LetK=⊕i∈IKi.(1)
The sequenceΛi∈EndA∗ℋ,Ki:i∈Iis a g-frame forℋwith respect toKii∈Iif and only if the sequenceΛi˜∈EndA∗ℋ,K:i∈Iis a g-frame forℋwith respect toK, withΛi˜x=…,0,0,Λix,0,0,…,∀x∈ℋ(2)
The sequenceΛi∈EndA∗ℋ,Ki:i∈Iis a∗-g-frame forℋwith respect toKii∈Iif and only if the sequenceΛi˜∈EndA∗ℋ,K:i∈Iis a∗-g-frame forℋwith respect toK, withΛi˜x=…,0,0,Λix,0,0,…,∀x∈ℋ(3)
ForK∈EndA∗ℋ, the sequenceΛi∈EndA∗ℋ,Ki:i∈Iis aK-g-frame forℋwith respect toKii∈Iif and only if the sequenceΛi˜∈EndA∗ℋ,K:i∈Iis aK-g-frame forℋwith respect toK, withΛi˜x=…,0,0,Λix,0,0,…,∀x∈ℋ(4)
ForK∈EndA∗ℋ, the sequenceΛi∈EndA∗ℋ,Ki:i∈Iis a∗-K-g-frame forℋwith respect toKii∈Iif and only if the sequenceΛi˜∈EndA∗ℋ,K:i∈Iis a∗-K-g-frame forℋwith respect toK, withΛi˜x=…,0,0,Λix,0,0,…,∀x∈ℋ


## 3. Frame forEndA∗ℋ,K

We begin this section with the following definition.Definition 10.
A sequenceTi∈EndA∗ℋ,K:i∈I is said to be a frame for EndA∗ℋ,K if there exist 0<A,B<∞ such that(24)AT,T≤∑i∈IT,TiTi,T≤T,T,∀T∈EndA∗ℋ,K,where the series converges in the strong operator topology.Example 3.
Forn∈ℕ∗, let Tii=1n⊂EndA∗ℋ,K such that, for all i=1,…,n,Ti is injective and has a closed range.
Then, for alli=1,…,n, we have(25)Ti∗Ti−1−1Iℋ≤Ti∗Ti≤Ti2Iℋ.
So,(26)∑i=1nTi∗Ti−1−1Iℋ≤∑i=1nTi∗Ti≤∑i=1nTi2Iℋ.
Hence,(27)∑i=1nTi∗Ti−1−1TT∗≤∑i=1nTTi∗TiT∗≤∑i=1nTi2TT∗,∀T∈EndA∗ℋ,K.
Thus,(28)∑i=1nTi∗Ti−1−1T,T≤∑i=1nT,TiTi,T≤∑i=1nTi2T,T,for all T∈EndA∗ℋ,K.
This shows thatTii=1n is a frame for EndA∗ℋ,K.Example 4.
LetTii∈ℕ∗⊂EndA∗ℋ,K such that, for all i∈ℕ∗,Ti is injective and has a closed range and Ti≤1/i.
Then, for alli∈ℕ∗, we have(29)Ti∗Ti−1−1Iℋ≤Ti∗Ti≤Ti2Iℋ.
So,(30)∑i∈ℕ∗Ti∗Ti−1−1Iℋ≤∑i∈ℕ∗Ti∗Ti≤∑i∈ℕ∗Ti2Iℋ.
Hence,(31)∑i∈ℕ∗Ti∗Ti−1−1TT∗≤∑i∈ℕ∗TTi∗TiT∗≤∑i∈ℕ∗Ti2TT∗,∀T∈EndA∗ℋ,K.
Thus,(32)∑i∈ℕ∗Ti∗Ti−1−1T,T≤∑i∈ℕ∗T,TiTi,T≤∑i∈ℕ∗Ti2T,T,for all T∈EndA∗ℋ,K.
This shows thatTii∈ℕ∗ is a frame for EndA∗ℋ,K.Theorem 2.
A sequenceTi∈EndA∗ℋ,K:i∈Iis a frame forEndA∗ℋ,Kif and only if it is a g-frame forℋwith respect toK.Proof.
LetTii∈I be a g-frame for ℋ with respect to K.
Then, there exist two positive constantsA and B, such that(33)Ax,xA≤∑i∈ITix,TixA≤Bx,xA,∀x∈ℋ,⟺Ax,xA≤∑i∈ITi∗Tix,xA≤Bx,xA,∀x∈ℋ.
So,(34)AIℋ≤∑i∈ITi∗Ti≤BIℋ.
Hence,(35)ATT∗≤∑i∈ITTi∗TiT∗≤BTT∗,∀T∈EndA∗ℋ,K.
Thus,(36)AT,T≤∑i∈IT,TiTi,T≤BT,T,∀T∈EndA∗ℋ,K,i.e., Tii∈I is a frame for EndA∗ℋ,K.
Conversely, assume thatTii∈I is a frame for EndA∗ℋ,K.
Then, there exist two positive constantsA and B, such that(37)AT,T≤∑i∈IT,TiTi,T≤BT,T,∀T∈EndA∗ℋ,K,⟺ATT∗≤∑i∈ITTi∗TiT∗≤BTT∗,∀T∈EndA∗ℋ,K.
So,(38)ATT∗x,xA≤∑i∈ITTi∗TiT∗x,xA≤BTT∗x,xA,∀T∈EndA∗ℋ,K,∀x∈K.
Lety∈ℋ and T∈EndA∗ℋ,K such that T∗x=y; then,(39)Ay,yA≤∑i∈ITi∗Tiy,yA≤By,yA,∀y∈ℋ,i.e.,(40)Ay,yA≤∑i∈ITiy,TiyA≤By,yA,∀y∈ℋ,and thus, Tii∈I is a g-frame for ℋ with respect to K.Corollary 4.
A sequenceTi∈EndA∗ℋ,K:i∈Iis a tight frame forEndA∗ℋ,Kif and only if it is a tight g-frame forℋwith respect toK.Corollary 5.
A sequenceTi∈EndA∗ℋ,K:i∈Iis a Bessel sequence forEndA∗ℋ,Kif and only if it is a g-Bessel sequence forℋwith respect toK.


## 4. Generalized Frame forEndA∗ℋ,K

We begin this section with the following definition.Definition 11.
A sequenceTi∈EndA∗ℋ,K:i∈I is said to be a generalized frame for EndA∗ℋ,K if there exist strictly positive elements A and B in A such that(41)AT,T≤∑i∈IT,TiTi,T≤BT,T,∀T∈EndA∗ℋ,K,where the series converges in the strong operator topology.
In the following,A is a commutative C∗-algebra.Theorem 3.
A sequenceTi∈EndA∗ℋ,K:i∈Iis a generalized frame forEndA∗ℋ,Kif and only if it is a∗-g-frame forℋwith respect toK.Proof.
LetTii∈I be a ∗-g-frame for ℋ with respect to K.
Then, there exist strictly positive elementsA and B in A, such that(42)Ax,xA≤∑i∈ITix,TixA≤Bx,xA,∀x∈ℋ,⟺Ax,xA≤∑i∈ITi∗Tix,xA≤Bx,xA,∀x∈ℋ.
So,(43)AIℋ≤∑i∈ITi∗Ti≤BIℋ.
Hence,(44)ATT∗≤∑i∈ITTi∗TiT∗≤BTT∗,∀T∈EndA∗ℋ,K.
Thus,(45)AT,T≤∑i∈IT,TiTi,T≤BT,T,∀T∈EndA∗ℋ,K,i.e., Tii∈I is a generalized frame for EndA∗ℋ,K.
Conversely, assume thatTii∈I is a generalized frame for EndA∗ℋ,K.
Then, there exist strictly positive elementsA and B in A, such that(46)AT,T≤∑i∈IT,TiTi,T≤BT,T,∀T∈EndA∗ℋ,K,⟺ATT∗≤∑i∈ITTi∗TiT∗≤BTT∗,∀T∈EndA∗ℋ,K.
So,(47)ATT∗x,xA≤∑i∈ITTi∗TiT∗x,xA≤BTT∗x,xA,∀T∈EndA∗ℋ,K,∀x∈K.
Lety∈ℋ and T∈EndA∗ℋ,K such that T∗x=y; then,(48)Ay,yA≤∑i∈ITi∗Tiy,yA≤By,yA,∀y∈ℋ,i.e.,(49)Ay,yA≤∑i∈ITiy,TiyA≤By,yA,∀y∈ℋ,and thus, Tii∈I is a ∗-g-frame for ℋ with respect to K.Corollary 6.
A sequenceTi∈EndA∗ℋ,K:i∈Iis a generalized tight frame forEndA∗ℋ,Kif and only if it is a tight∗-g-frame forℋwith respect toK.Corollary 7.
A sequenceTi∈EndA∗ℋ,K:i∈Iis a generalized Bessel sequence forEndA∗ℋ,Kif and only if it is a∗-g-Bessel sequence forℋwith respect toK.


## 5. K-Frame forEndA∗ℋ,K

We begin this section with the following definition.Definition 12.
LetK∈EndA∗ℋ. A sequence Ti∈EndA∗ℋ,K:i∈I is said to be a K-frame for EndA∗ℋ,K if there exist 0<A,B<∞ such that(50)ATK,TK≤∑i∈IT,TiTi,T≤BT,T,∀T∈EndA∗ℋ,K,where the series converges in the strong operator topology.Example 5.
Forn∈ℕ∗, let Tii=1n⊂EndA∗ℋ.
Then, forK=∑i=1nTi∗Ti1/2, we have(51)K∗K=∑i=1nTi∗Ti.
So,(52)TKK∗T∗=∑i=1nTTi∗TiT∗.
Hence,(53)TK,TK=∑i=1nT,TiTi,T,∀T∈EndA∗ℋ.
This shows thatTii=1n is a Parseval K-frame for EndA∗ℋ.
Similar to Remark 1 in [22], we have the following.Remark 1.
(1)
Every frame forEndA∗ℋ,K is a K-frame, for any K∈EndA∗ℋ: K≠0(2)
IfK∈EndA∗ℋ is a surjective operator, then every K-frame for EndA∗ℋ,K is a frame for EndA∗ℋ,KTheorem 4.
LetK∈EndA∗ℋandTi∈EndA∗ℋ,K:i∈Ibe a frame forEndA∗ℋ,K. Then,TiK∗i∈Iis aK-frame forEndA∗ℋ,K.Proof.
LetTii∈I be a frame for EndA∗ℋ,K.
Then, there exist two positive constantsA and B, such that(54)AT,T≤∑i∈IT,TiTi,T≤BT,T,∀T∈EndA∗ℋ,K.
Hence,(55)ATK,TK≤∑i∈ITK,TiTi,TK≤BTK,TK,∀T∈EndA∗ℋ,K.
Thus,(56)ATK,TK≤∑i∈IT,TiK∗TiK∗,T≤BK2T,T,∀T∈EndA∗ℋ,K.
Then,TiK∗i∈I is a K-frame for EndA∗ℋ,K.Theorem 5.
LetK1,K2∈EndA∗ℋandTi∈EndA∗ℋ,K:i∈Ibe aK1-frame forEndA∗ℋ,K. Then,TiK∗i∈Iis aK2K1-frame forEndA∗ℋ,K.Proof.
LetTii∈I be a K1-frame for EndA∗ℋ,K.
Then, there exist two positive constantsA and B, such that(57)ATK1,TK1≤∑i∈IT,TiTi,T≤BT,T,∀T∈EndA∗ℋ,K.
Hence,(58)ATK2K1,TK2K1≤∑i∈ITK2,TiTi,TK2≤BTK2,TK2,∀T∈EndA∗ℋ,K.
Thus,(59)ATK2K1,TK2K1≤∑i∈IT,TiK2∗TiK2∗,T≤BK22T,T,∀T∈EndA∗ℋ,K.
Then,TiK∗i∈I is a K2K1-frame for EndA∗ℋ,K.Corollary 8.
LetK∈EndA∗ℋandTi∈EndA∗ℋ,K:i∈Ibe aK-frame forEndA∗ℋ,K. Then,TiK∗Ni∈Iis aKN+1-frame forEndA∗ℋ,K.Proof.
It follows from the previous theorem.Theorem 6.
A sequenceTi∈EndA∗ℋ,K:i∈Iis aK-frame forEndA∗ℋ,Kif and only if it is aK-g-frame forℋwith respect toK.Proof.
LetTii∈I be a K-g-frame for ℋ with respect to K.
Then, there exist two positive constantsA and B, such that(60)AK∗x,K∗xA≤∑i∈ITix,TixA≤Bx,xA,∀x∈ℋ,⟺AKK∗x,xA≤∑i∈ITi∗Tix,xA≤Bx,xA,∀x∈ℋ.
So,(61)AKK∗≤∑i∈ITi∗Ti≤BIℋ.
Hence,(62)ATKK∗T∗≤∑i∈ITTi∗TiT∗≤BTT∗,∀T∈EndA∗ℋ,K.
Thus,(63)ATK,TK≤∑i∈IT,TiTi,T≤BT,T,∀T∈EndA∗ℋ,K,i.e., Tii∈I is a K-frame for EndA∗ℋ,K.
Conversely, assume thatTii∈I is a K-frame for EndA∗ℋ,K.
Then, there exist two positive constantsA and B, such that(64)ATK,TK≤∑i∈IT,TiTi,T≤BT,T,∀T∈EndA∗ℋ,K,⟺ATKK∗T∗≤∑i∈ITTi∗TiT∗≤BTT∗,∀T∈EndA∗ℋ,K.
So,(65)ATKK∗T∗x,xA≤∑i∈ITTi∗TiT∗x,xA≤BTT∗x,xA,∀T∈EndA∗ℋ,K,∀x∈K.
Lety∈ℋ and T∈EndA∗ℋ,K such that T∗x=y; then,(66)AKK∗y,yA≤∑i∈ITi∗Tiy,yA≤By,yA,∀y∈ℋ,i.e.,(67)AK∗y,K∗yA≤∑i∈ITiy,TiyA≤By,yA,∀y∈ℋ,and thus, Tii∈I is a K-g-frame for ℋ with respect to K.Corollary 9.
A sequenceTi∈EndA∗ℋ,K:i∈Iis a tightK-frame forEndA∗ℋ,Kif and only if it is a tightK-g-frame forℋwith respect toK.


## 6. Generalized K-Frame forEndA∗ℋ,K

We begin this section with the following definition.Definition 13.
LetK∈EndA∗ℋ. A sequence Ti∈EndA∗ℋ,K:i∈I is said to be a generalized K-frame for EndA∗ℋ,K if there exist strictly positive elements A and B in A such that(68)ATK,TK≤∑i∈IT,TiTi,T≤BT,T,∀T∈EndA∗ℋ,K,where the series converges in the strong operator topology.
In the following,A is a commutative C∗-algebra.Theorem 7.
A sequenceTi∈EndA∗ℋ,K:i∈Iis a generalizedK-frame forEndA∗ℋ,Kif and only if it is a∗-K-g-frame forℋwith respect toK.Proof.
LetTii∈Ibe an∗-K-g-frame forℋwith respect toK.Then, there exist strictly positive elementsA and B in A, such that(69)AK∗x,K∗xA≤∑i∈ITix,TixA≤Bx,xA,∀x∈ℋ,⟺AKK∗x,xA≤∑i∈ITi∗Tix,xA≤Bx,xA,∀x∈ℋ.So,(70)AKK∗≤∑i∈ITi∗Ti≤BIℋ.Hence,(71)ATKK∗T∗≤∑i∈ITTi∗TiT∗≤BTT∗,∀T∈EndA∗ℋ,K.Thus,(72)ATK,TK≤∑i∈IT,TiTi,T≤BT,T,∀T∈EndA∗ℋ,K,i.e.,Tii∈I is a generalized K-frame for EndA∗ℋ,K.Conversely, assume thatTii∈I is a generalized K-frame for EndA∗ℋ,K.Then, there exist strictly positive elementsA and B in A, such that(73)ATK,TK≤∑i∈IT,TiTi,T≤BT,T,∀T∈EndA∗ℋ,K,⟺ATKK∗T∗≤∑i∈ITTi∗TiT∗≤BTT∗,∀T∈EndA∗ℋ,K.So,(74)ATKK∗T∗x,xA≤∑i∈ITTi∗TiT∗x,xA≤BTT∗x,xA,∀T∈EndA∗ℋ,K,∀x∈K.Lety∈ℋ and T∈EndA∗ℋ,K such that T∗x=y; then,(75)AKK∗y,yA≤∑i∈ITi∗Tiy,yA≤By,yA,∀y∈ℋ,i.e.,(76)AK∗y,K∗yA≤∑i∈ITiy,TiyA≤By,yA,∀y∈ℋ,and thus, Tii∈I is a ∗-K-g-frame for ℋ with respect to K.Corollary 10.
A sequenceTi∈EndA∗ℋ,K:i∈Iis a generalized tightK-frame forEndA∗ℋ,Kif and only if it is a tight∗-K-g-frame forℋwith respect toK.



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*Source: 5522671-2021-05-22.xml*