# Perturbation and Stability of Continuous Operator Frames in HilbertC∗-Modules

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

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## Abstract

Frame theory has a great revolution in recent years. This theory has been extended from the Hilbert spaces to HilbertC∗-modules. In this paper, we consider the stability of continuous operator frame and continuous K-operator frames in Hilbert C∗-modules under perturbation, and we establish some properties.


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## Body

## 1. Introduction and Preliminaries

The concept of frames in Hilbert spaces is a new theory which was introduced by Duffin and Schaeffer [1] in 1952 to study some deep problems in nonharmonic Fourier series. This theory was reintroduced and developed by Daubechieset al. [2].In 1993, Aliet al. [3] introduced the concept of continuous frames in Hilbert spaces. Gabardo and Han in [4] called these kinds of frames, frames associated with measurable spaces.In 2000, Frank and Larson [5] introduced the notion of frames in Hilbert C∗-modules as a generalization of frames in Hilbert spaces. The theory of continuous frames has been generalized in Hilbert C∗-modules. For more details, see [6–25].The aim of this paper is to extend the results of Rossafi and Akhlidj [23], given for Hilbert C∗-module in a discrete case.In the following, we briefly recall the definitions and basic properties ofC∗-algebra and Hilbert A-modules. Our references for C∗-algebras are [26, 27]. For C∗-algebra A, if a∈A is positive, we write a≥0, and A+ denotes the set of positive elements of A.Definition 1.
(see [26]). Let A be unital C∗-algebra and ℋ be 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 it is sesquilinear and 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 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.
The following lemmas will be used to prove our mains results.Lemma 1 (see [28]).
Letℋ be a Hilbert A-module. If T∈EndA∗ℋ, then(1)Tx,TxA≤T2x,xA,x∈ℋ.Lemma 2 (see [29]).
Letℋ and K be two Hilbert A-modules and T∈EndA∗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 τ>0 such that T∗x≥τx, for all x∈K.(iii)
T∗ is bounded below with respect to the inner product, i.e., there is ζ>0 such that T∗x,T∗xA≥ζx,xA, for all x∈K.Lemma 3 (see [30]).
LetΩ,μ be a measure space, X and Y are two Banach spaces, λ:X⟶Y be a bounded linear operator and f:Ω⟶X measurable function; then,(2)λ∫Ωfdμ=∫Ωλfdμ.


## 2. Characterisation of Continuous Operator Frame forEndA∗ℋ

LetX be a Banach space, Ω,μ a measure space, and f:Ω⟶X be a measurable function. Integral of Banach-valued function f has been defined by Bochner and others. Most properties of this integral are similar to those of the integral of real-valued functions [30, 31]. Since every C∗-algebra and Hilbert C∗-module are Banach spaces, we can use this integral and its properties.LetΩ,μ be a measure space, U and V be two Hilbert C∗-modules over a unital C∗-algebra and Vww∈Ω is a family of submodules of V. EndA∗U,Vw is the collection of all adjointable A-linear maps from U into Vw.We define the following:(3)l2Ω,Vwω∈Ω=x=xww∈Ω:xw∈Vw,∫Ωxw2dμw<∞.For anyx=xww∈Ω and y=yww∈Ω, the A-valued inner product is defined by x,y=∫Ωxw,ywAdμw and the norm is defined by x=x,x1/2. In this case, l2Ω,Vwω∈Ω is an Hilbert C∗-module [32].Definition 2.
We callΛ:=Λw∈EndA∗ℋ:w∈Ω a continuous operator frame for EndA∗ℋ if(a)
for anyx∈ℋ, the mapping x˜:Ω⟶Vw defined by x˜w=Λwx is measurable(b)
there is a pair of constants0<ν,δ such that for any x∈ℋ,(4)νx,xA≤∫ΩΛwx,ΛwxAdμw≤δx,xA,x∈ℋ.
The constantsν and δ are called continuous operator frame bounds.
Ifν=δ, we call this continuous operator frame a continuous tight operator frame, and if ν=δ=1, it is called a continuous Parseval operator frame.
If only the right-hand inequality of (4) is satisfied, we call Λ=Λww∈Ω the continuous Bessel operator frame for EndA∗ℋ with Bessel bound δ.
The continuous frame operatorS of Λ on ℋ is defined by(5)Sx=∫ΩΛω∗Λωxdμω,x∈ℋ.
The continuous frame operator S is a bounded, positive, self-adjoint, and invertible.Theorem 1.
LetΛ=Λw∈EndA∗ℋ:w∈Ω. Λ is a continuous operator frame for EndA∗ℋ if and only if there exist constants 0<ν,δ such that for any x∈ℋ,(6)νx2≤∫ΩΛwx,ΛwxAdμw≤δx2.


## 3. Perturbation and Stability of Continuous Operator Frame forEndA∗ℋ

Theorem 2.
LetTww∈Ω be a continuous operator frame for EndA∗ℋ with bounds ν and δ. If Rww∈Ω⊂EndA∗ℋ is a continuous operator Bessel family with bound ξ<ν, then Tw∓Rww∈Ω is a continuous operator frame for EndA∗ℋ.Proof.
We just prove the case thatTw+Rww∈Ω is a continuous operator frame for EndA∗ℋ.
On the one hand, for eachx∈ℋ, we have(7)Tw+Rwxw∈Ω=∫ΩTw+Rwx,Tw+RwxAdμw1/2≤Twxw∈Ω+Rwxw∈Ω=∫ΩTwx,TwxAdμw1/2+∫ΩRwx,RwxAdμw1/2≤δx+ξx.
Hence,(8)∫ΩTw+Rwx,Tw+RwxAdμw1/2≤δ+ξx.
One the other hand, we have(9)Tw+Rwxw∈Ω=∫ΩTw+Rwx,Tw+RwxAdμw1/2≥Twxw∈Ω−Rwxw∈Ω=∫ΩTwx,TwxAdμw1/2−∫ΩRwx,RwxAdμw1/2≥νx−ξx.
Then,(10)∫ΩTw+Rwx,Tw+RwxAdμw1/2≥ν−ξx.
From (8) and (9), we get(11)ν−ξ2x2≤∫ΩTw+Rwx,Tw+RwxAdμw≤δ+ξ2x2.
Therefore,Tw+Rww∈Ω is a continuous operator frame for EndA∗ℋ.Theorem 3.
LetTww∈Ω be a continuous operator frame for EndA∗ℋ with bounds ν and δ and let Rww∈Ω⊂EndA∗ℋ. The following statements are equivalent:(i)
Rww∈Ω is a continuous operator frame for EndA∗ℋ.(ii)
There exists a constantξ>0, such that for all x in ℋ, we have(12)∫ΩTw−Rwx,Tw−RwxAdμw≤ξ.min∫ΩTwx,TwxAdμw,∫ΩRwx,RwxAdμw.Proof.
Suppose thatRww∈Ω is a continuous operator frame for EndA∗ℋ with bound η and ρ. Then for all x∈ℋ, we have(13)Tw−Rwxw∈Ω=∫ΩTw−Rwx,Tw−RwxAdμw1/2≤Twxw∈Ω+Rwxw∈Ω=∫ΩTwx,TwxAdμw1/2+∫ΩRwx,RwxAdμw1/2≤∫ΩTwx,TwxAdμw1/2+ρx≤∫ΩTwx,TwxAdμw1/2+ρν∫ΩTwx,TwxAdμw1/2=1+ρν∫ΩTwx,TwxAdμw1/2.
In the same way, we have(14)∫ΩTw−Rwx,Tw−RwxAdμw1/2≤1+δη∫ΩRwx,RwxAdμw1/2.
For (12), we take ξ=min1+δ/η,1+ρ/ν.
Now we assume that (12) holds. For each x∈ℋ, we have(15)νx≤∫ΩTwx,TwxAdμw1/2=Twxw∈Ω≤Tw−Rwxw∈Ω+Rwxw∈Ω=∫ΩTw−Rwx,Tw−RwxAdμw1/2+∫ΩRwx,RwxAdμw1/2.
From (12), we have(16)∫ΩTw−Rwx,Tw−RwxAdμw≤ξ∫ΩRwx,RwxAdμw.
Then,(17)∫ΩTwx,TwxAdμw1/2≤ξ∫ΩRwx,RwxAdμw1/2+∫ΩRwx,RwxAdμw1/2.
Hence,(18)νx≤∫ΩTwx,TwxAdμw1/2≤1+ξ∫ΩRwx,RwxAdμw1/2.
Also, we have(19)Rwxw∈Ω=∫ΩRwx,RwxAdμw1/2=Rwx−Twx+Twxw∈Ω‖Tw−Rwxw∈Ω+Twxw∈Ω=∫ΩTw−Rwx,Tw−RwxAdμw1/2+∫ΩTwx,TwxAdμw1/2.
From (12), we have(20)∫ΩTw−Rwx,Tw−RwxAdμw≤ξ∫ΩTwx,TwxAdμw.
Then,(21)∫ΩRwx,RwxAdμw1/2≤1+ξ∫ΩTwx,TwxAdμw1/2.
So,(22)∫ΩRwx,RwxAdμw1/2≤1+ξδx.
From (18) and (22), we give that(23)ν1+ξ2x2≤∫ΩRwx,RwxAdμw≤δ1+ξ2x2.
Therefore,Rww∈Ω is a continuous operator frame for EndA∗ℋ.Theorem 4.
LetTk,ww∈Ω⊂EndA∗ℋ, k=1,2,…,n be a continuous operator frames for EndA∗ℋ with bounds νk and δk and let αkk∈1,…,n be any scalars. If there exists a constant λ>0 and some p∈1,2,…,n such that(24)λTp,ww∈Ω≤∑k=1nαkTk,wx,x∈ℋ,then ∑k=1nαkTk,ww∈Ω is a continuous operator frame for EndA∗ℋ and conversely.Proof.
For everyx∈ℋ, we have(25)νpλx,xA1/2≤Tp,wxw∈Ω≤∑k=1nαkTk,wxw∈Ω≤∑k=1nαkTk,wxw∈Ω≤max1≤k≤nαk∑k=1nTk,wxw∈Ω≤max1≤k≤nαk∑k=1nδkx,xA1/2.
Hence(26)νpλ2x,xA≤∑k=1nαkTk,wxw∈Ω2≤max1≤k≤nαk2∑k=1nδk2x,xA.
Therefore,∑k=1nαkTk,wxw∈Ω is a continuous operator frame for EndA∗ℋ.
For the converse, let∑k=1nαkTk,wxw∈Ω be a continuous operator frame for EndA∗ℋ with bounds ν, δ and let any k∈1,2,…,n.
SinceTp,ww∈Ω is a continuous operator frame for EndA∗ℋ with bounds νp and δp, then for any x∈ℋ, we have(27)νpx,xA≤Tp,ww∈Ω2≤δpx,xA.
Hence,(28)δp−1Tp,ww∈Ω2≤x,xA.
Also, we have(29)νx,xA≤∑k=1nαkTk,wxw∈Ω2,x∈ℋ.
Then,(30)x,xA≤ν−1∑k=1nαkTk,wxw∈Ω2,x∈ℋ.
So,(31)νδpTp,ww∈Ω2≤∑k=1nαkTk,wxw∈Ω2,x∈ℋ.
Then forλ=ν/δp, we have(32)λTp,ww∈Ω2≤∑k=1nαkTk,wxw∈Ω2,x∈ℋ,which ends the proof.Theorem 5.
Fork=1,2,…,n, let Tk,ww∈Ω⊂EndA∗ℋ be continuous operator frames with bounds νk and δk and let Rk,ww∈Ω⊂EndA∗ℋ.
LetL:l2ℋ⟶l2ℋ be a bounded linear operator such that(33)L∑k=1nRk,ww∈Ω=Tp,ww∈Ω,forsomep∈1,2,…,n.
If there exists a constantλ>0 such that for each x∈ℋ and k=1,…,n, we have(34)∫ΩTk,w−Rk,wx,Tk,w−Rk,wxAdμw≤λ∫ΩTwx,TwxAdμw.
Then∑k=1nRk,ww∈Ω is a continuous operator frame for EndA∗ℋ.Proof.
For allx∈ℋ, we have(35)∑k=1nRk,wxw∈Ω≤∑k=1nRk,wxw∈Ω≤∑k=1nTk,w−Rk,wxw∈Ω+Tk,wxw∈Ω≤1+λ∑k=1nTk,wxw∈Ω≤1+λ∑k=1nδkx,xA1/2.
Since, for anyx∈ℋ, we have(36)L∑k=1nRk,ww∈Ω=Tp,ww∈Ω.
Then(37)νpx,xA1/2≤Tp,ww∈Ω=L∑k=1nRk,ww∈Ω≤L∑k=1nRk,ww∈Ω.
Hence(38)νpLx,xA1/2≤∑k=1nRk,ww∈Ω,x∈ℋ.
Therefore(39)νpLx,xA1/2≤∑k=1nRk,ww∈Ω≤1+λ∑k=1nδkx,xA1/2.
This give that∑k=1nRk,ww∈Ω is a continuous operator frame for EndA∗ℋ.


## 4. Characterisation of Continuous K-Operator Frames forEndA∗ℋ

Definition 3.
LetK∈EndA∗ℋ. A family of adjointable operators Tww∈Ω on a Hilbert A-module ℋ is said to be a continuous K-operator frame for EndA∗ℋ, if there exists two positive constants ν,δ>0 such that(40)δK∗x,K∗xA≤∫ΩTwx,TwxAdμw≤δx,xA,x∈ℋ.
The numbersν and δ are called, respectively, lower and upper bound of the continuous K-operator frame.
The continuous K-operator frame is called aν-thight if:(41)νK∗x,K∗xA=∫ΩTwx,TwxAdμw.
Ifν=1, it is called a normalised tight continuous K-operator frame or a Parseval continuous K-operator frame. The continuous K-operator frame is standard if for every x∈ℋ, the sum (40) converges in norm.Remark 1.
For anyK∈EndA∗ℋ, every continuous operator frame is a continuous K-operator frame.
Indeed, for anyK∈EndA∗ℋ, we have(42)K∗x,K∗xA≤K2x,xA,x∈ℋ.
LetTww∈Ω be a continuous operator frame with bounds ν and δ, then(43)νK−2K∗x,K∗xA≤νx,xA≤∫ΩTwx,TwxAdμw≤δx,xA,x∈ℋ.
HenceTww∈Ω is a continuous K-operator frame with bounds νK−2 and δ.
LetTww∈Ω be a continuous K-operator for EndA∗ℋ. We define the operator(44)ℛ:ℋ⟶l2ℋ,x⟶ℛx=Twxw∈Ω.
The operatorℛ is called the analysis operator of the continuous K-operator frame Tww∈Ω, and its adjoint is defined as follows:(45)ℛ∗:l2ℋ⟶ℋ,xww∈Ω⟶ℛ∗xww∈Ω=∫ΩTw∗xwdμω.
The operatorsℛ is called the synthesis operator of the continuous K-operator frame Tww∈Ω.
By composingℛ and ℛ∗, we obtain the operator(46)SK:ℋ⟶ℋx⟶SKx=ℛ∗ℛx=∫ΩTw∗Twxdμω.
It is easy to show that the operatorSK is positive and self-adjoint.Theorem 6.
LetTww∈Ω be a family of adjointable operators on a Hilbert A-module ℋ. Assume that ∫ΩTwx,TwxAdμw converges in norm for all x∈ℋ. Then Tww∈Ω is a continuous K-operator frame for EndA∗ℋ if and only if there exists two positive constants ν,δ>0 such that(47)νK∗x2≤∫ΩTwx,TwxAdμw≤δx2.Proof.
Suppose thatTww∈Ω is a continuous K-operator frame.
From the definition of continuous K-operator frame, (47) holds.
Conversely, assume that (47) holds. The frame operator SK is positive and self-adjoint; then(48)SK1/2x,SK1/2xA=SKx,xA=∫ΩTwx,TwxAdμw.
We have for any(49)x∈ℋ,νK∗x≤SK1/2x≤δx.
Using Lemma2, there exist two constants τ,ξ>0 such that(50)τK∗x,K∗xA≤SK1/2x,SK1/2xA=∫ΩTwx,TwxAdμω≤ξx,xA.
This proves thatTww∈Ω is a continuous K-operator frame for EndA∗ℋ.


## 5. Perturbation and Stability of Continuous K-Operator Frames forEndA∗ℋ

Theorem 7.
LetTww∈Ω be a continuous K-operator frame for EndA∗ℋ with bounds A and B, let Rww∈Ω⊂EndA∗ℋ and αww∈Ω,βww∈Ω∈ℝ be two positively families. If there exist two constants 0≤λ,μ<1 such that for any x∈ℋ, we have(51)∫ΩαwTw−βwRwx,αwTw−βwRwxAdμω1/2≤λ∫ΩαwTwx,αwTwxAdμω1/2+μ∫ΩβwRwx,βwRwxAdμω1/2.ThenRww∈Ω is a continuous K-operator frame for EndA∗ℋ.Proof.
For everyx∈ℋ, we have(52)βwRwxw∈Ω≤αwTw−βwRwxw∈Ω+αwTwxw∈Ω≤μβwRwxw∈Ω+λαwTwxw∈Ω+αwTwxw∈Ω=1+λαwTwxw∈Ω+μβwRwxw∈Ω.
Then,(53)1−μβwRwxw∈Ω≤1+λαwTwx.
Therefore(54)1−μinfω∈ΩβwRwxw∈Ω≤1+λsupω∈ΩαwTwxw∈Ω.
Hence(55)Rwxw∈Ω≤1+λsupω∈Ωαw1−μinfω∈ΩβwTwxw∈Ω.
Also, for allx∈ℋ, we have(56)αwTwxw∈Ω≤αwTw−βwRwxw∈Ω+βwRwxw∈Ω≤μβwRwxw∈Ω+λαwTwxw∈Ω+αwTwxw∈Ω=λαwTwxw∈Ω+1+μβwRwxw∈Ω,then(57)1−λαwTwxw∈Ω≤1+μβwRwxw∈Ω.
Hence(58)1−λinfω∈ΩαwTwxw∈Ω≤1+μsupω∈ΩβwRwxw∈Ω.
Thus(59)1−λinfω∈Ωαw1+μsupω∈ΩβwTwxw∈Ω≤Rwxw∈Ω.
Therefore(60)ν1−λinfω∈Ωαw1+μsupω∈Ωβw2x,xA≤1−λinfω∈Ωαw1+μsupω∈Ωβw2Twxw2≤Rwxw2.
So,(61)Rwxw∈Ω2≤1+λsupω∈Ωαw1−μinfω∈Ωβw2Twxw∈Ω2≤δ1+λsupαw1−μinfω∈Ωβw2x,xA.
Hence(62)ν1−λinfω∈Ωαw1+μsupω∈Ωβw2x,xA≤∫ΩRwx,RwxAdμω≤δ1+λsupω∈Ωαw1−μinfω∈Ωβw2x,xA.
This give thatRww∈Ω is a continuous K-operator frame for EndA∗ℋ.Theorem 8.
LetTww∈Ω be a continuous K-operator frame for EndA∗ℋ with bounds ν and δ. Let Rww∈Ω∈EndA∗ℋ and . If 0≤α+β/ν<1 such that for all x∈ℋ, we have(63)∫ΩTw−Rwx,Tw−RwxAdμω≤α∫ΩTwx,TwxAdμω+βK∗x,K∗xA.
ThenRww∈Ω is a continuous K-operator frame with bounds ν1−α+β/ν2 and δ1+α+β/ν2.Proof.
LetTww∈Ω be a continuous K-operator frame with bounds ν and δ. Then for any x∈ℋ, we have(64)Twxw∈Ω≤Tw−Rwxw∈Ω+Rwxw∈Ω≤α∫ΩTwx,TwxAdμω+βK∗x,K∗xA1/2+∫ΩRwx,RwxAdμω1/2≤α∫ΩTwx,TwxAdμω+βν∫ΩTwx,TwxAdμω1/2+∫ΩRwx,RwxAdμω1/2=α+βνTwxw∈Ω+∫ΩRwx,RwxAdμω1/2.
Therefore(65)1−α+βνTwxw∈Ω≤∫ΩRwx,RwxAdμω1/2.
Thus(66)ν1−α+βν2K∗x,K∗xA≤1−α+βν2∫ΩTwx,TwxAdμω≤∫ΩRwx,RwxAdμω,ν1−α+βν2K∗x,K∗xA≤1−α+βν2∫ΩTwx,TwxAdμω∫ΩRwx,RwxAdμω.
Also, we have(67)Rwxw∈Ω≤Tw−Rwxw∈Ω+Twxw∈Ω≤α+βνTwxw∈Ω+Twxw∈Ω=1+α+βνTwxw∈Ω≤δ1+α+βνx,xA1/2.
Hence(68)∫ΩRwx,RwxAdμω≤δ1+α+βν2x,xA.
Therefore(69)ν1−α+βν2K∗x,K∗xA≤∫ΩRwx,RwxAdμω≤δ1+α+βν2x,xA.
HenceRww∈Ω is a continuous K-operator frame with bounds ν1+α−β/ν2 and δ1+α+β/ν2.Corollary 1.
LetTww∈Ω be a continuous K-operator frame for EndA∗ℋ with bounds ν and δ. Let Rww∈Ω⊂EndA∗ℋ and 0≤α. If 0≤α<ν such that(70)∫ΩTw−Rwx,Tw−RwxAdμω≤αK∗x,K∗xA,x∈ℋ,then Rww∈Ω is a continuous K-operator frame with bounds ν1−α/ν2 and δ1+α/ν2.Proof.
The proof comes from the previous theorem.Theorem 9.
LetTww∈Ω be a continuous K-operator frame for EndA∗ℋ with bounds ν and δ. Let Rww∈Ω⊂EndA∗ℋ. If there exists ξ>0 such that for any x∈ℋ, we have(71)∫ΩTw−Rwx,Tw−RwxAdμω≤ξmin∫ΩRwx,RwxAdμω,∫ΩRwx,RwxAdμω.
ThenRww∈Ω is a continuous K-operator frame for EndA∗ℋ. The converse is true for any surjective operator K such that   in particular if K is co-isometry.Proof.
Assume that (71) holds.
On the one hand, we have for anyx in ℋ(72)νK∗x≤∫ΩTwx,TwxAdμω1/2=Twxw∈Ω≤Tw−Rwx+Rwx=∫ΩTw−Rwx,Tw−RwxAdμω1/2+∫ΩRwx,RwxAdμω1/2≤ξ∫ΩRwx,RwxAdμω1/2+∫ΩRwx,RwxAdμω1/2=1+ξ∫ΩRwx,RwxAdμω1/2.(73)νK∗x≤1+ξ∫ΩRwx,RwxAdμω1/2.
On the other hand, we have(74)∫ΩRwx,RwxAdμω1/2=Rwxw∈Ω≤Twxw∈Ω+Tw−Rwxw∈Ω=∫ΩTwx,TwxAdμω1/2+∫ΩTw−Rwx,Tw−RwxAdμω1/2≤∫ΩTwx,TwxAdμω1/2+ξ∫ΩTwx,TwxAdμω1/2=1+ξ∫ΩTwx,TwxAdμω1/2≤δ1+ξx,xA.
Then(75)∫ΩRwx,RwxAdμω1/2≤δ1+ξx,xA.
From (73) and (75), we obtain(76)ν1+ξ2K∗x2≤∫ΩRwx,RwxAdμω≤δ1+ξ2x,xA2.
HenceRww∈Ω is a continuous K-operator frame for EndA∗ℋ.
For the converse, ifRww∈Ω is a continuous K-operator frame for EndA∗ℋ with bound η and ρ, and K verify that,  i.e, x≤K∗x, then for every x∈ℋ, we have(77)Tw−Rwxw∈Ω=∫ΩTw−Rwx,Tw−RwxAdμω1/2≤Twxw∈Ω+Rwxw∈Ω=∫ΩTwx,TwxAdμω1/2+∫ΩRwx,RwxAdμω1/2≤∫ΩTwx,TwxAdμω1/2+ρx≤∫ΩTwx,TwxAdμω1/2+ρK∗x≤∫ΩTwx,TwxAdμω1/2+ρν∫ΩTwx,TwxAdμω1/2=1+ρν∫ΩTwx,TwxAdμω1/2.
Similarly we can obtain(78)∫ΩTw−Rwx,Tw−RwxAdμω1/2≤1+δη∫ΩRwx,RwxAdμω1/2.
We takeξ=min1+ρ/ν,1+δ/η, then (5.1) is verified.Theorem 10.
LetK∈EndA∗ℋ. For k=1,2,…,n, let Tk,ww∈Ω⊂EndA∗ℋ be a continuous K-operator frame for EndA∗ℋ with bounds νk and δk, {αk} be any scalars. If there exists a constant λ>0 and p∈1,2,…,n such that(79)λTp,wxw≤∑k=1nαkTk,wxw,x∈ℋ,then ∑k=1nαkTk,wxw∈Ω is a continuous K-operator frame for EndA∗ℋ. The converse is true for every co-isometric operator K.Proof.
For allx∈ℋ, we have(80)νpλK∗x,K∗xA1/2≤λTp,wxw∈Ω≤∑k=1nαkTk,wxw∈Ω≤∑k=1nαkTk,wxw∈Ω≤max1≤k≤nαk∑k=1nTk,wxw∈Ω≤max1≤k≤nαk∑k=1nδkx,A1/2.
Hence for anyx∈ℋ, we have(81)νpλK∗x,K∗xA1/2≤∑k=1nαkTk,wxw∈Ω≤max1≤k≤nαk∑k=1nδkx,xA1/2.
Then(82)νpλ2K∗x,K∗xA≤∑k=1nαkTk,wxw∈Ω2≤max1≤k≤nαk2∑k=1nδk2x,xA.
This gives that∑k=1nαkTk,wxw∈Ω is a continuous K-operator frame for EndA∗ℋ. For the converse, let K be a co-isometric operator on ℋ, let ∑k=1nαkTk,wxw is a continuous K-operator frame for EndA∗ℋ with bounds ν and δ and let for all p∈1,2,…,n, Tk,wxw be a continuous K-operator frame for EndA∗ℋ with bounds νp and δp. Then, for every x∈ℋ, we have(83)νpK∗x,K∗xA≤Tp,wxw∈Ω2≤δpx,xA.
Then(84)1δpTk,wxw∈Ω2≤x,xA.
Also, we have(85)νK∗x,K∗xA≤∑k=1nαkTk,wxw∈Ω2≤δx,xA.
SinceK is a co-isometric operator, then(86)x,xA=K∗x,K∗xA≤1ν∑k=1nαkTk,wxw∈Ω2,x∈ℋ.
So(87)νδpTp,wxw2≤∑k=1nαkTk,wxw2,x∈ℋ.
Therefore, forλ=ν/δp, we have(88)λTp,wxw∈Ω≤∑k=1nαkTk,wxw∈Ω,x∈ℋ.Theorem 11.
LetK∈EndA∗ℋ. For k=1,2,…,n, let Tk,ww∈Ω⊂EndA∗ℋ be a continuous K-operator frame for EndA∗ℋ with bounds νk and δk, and let Rk,w⊂EndA∗ℋ be any family. Let S: l2ℋ⟶l2ℋ be a bounded linear operator such that S∑k=1nRk,wxw∈Ω=Tp,wxw∈Ω, for some p∈1,2,…,n. If there exists a constant λ>0 such that(89)∫ΩTk,w−Rk,wx,Tk,w−Rk,wxAdμω≤λ∫ΩTwx,TwxAdμω,x∈ℋ,k=1,2,…,n,then ∑k=1nRk,ww∈Ω is a continuous K-operator frame for EndA∗ℋ.Proof.
For everyx∈ℋ, we have(90)∑k=1nRk,wxw∈Ω≤∑k=1nRk,wxw∈Ω≤∑k=1nTk,w-Rk,wxw∈Ω+Tk,wxw∈Ω≤λTk,wx+Tk,wxw∈Ω=1+λ∑k=1nTk,wx≤1+λ∑k=1nδkx,xA1/2.
We have for any(91)x∈ℋ,S∑k=1nRk,wxw∈Ω=Tp,wxw∈Ω.
Hence(92)νpK∗x,K∗xA1/2≤Tp,wxw∈Ω=S∑k=1nRk,wxw∈Ω≤S.∑k=1nRk,wxw∈Ω,x∈H.
So(93)νpSK∗x,K∗xA1/2≤∑k=1nRk,wxw∈Ω,x∈ℋ.
Therefore(94)νpSK∗x,K∗xA1/2≤∑k=1nRk,wxw∈Ω≤1+λ∑k=1nδkx,xA1/2,x∈ℋ.
Thus(95)νpS2K∗x,K∗xA≤∑k=1nRk,wxw∈Ω2≤1+λ2∑k=1nδk2x,xA,x∈ℋ.
This gives that∑k=1nRk,ww∈Ω is a continuous K-operator frame for EndA∗ℋ.



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*Source: 5576534-2021-06-16.xml*