# Controlled Continuous∗-K-g-Frames for Hilbert C∗-Modules

Authors: Abdeslam Touri; Hatim Labrigui; Mohamed Rossafi; Samir Kabbaj
Journal: Abstract and Applied Analysis
 (2021)

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## Abstract

Frame theory has a great revolution for recent years. This theory has been extended from Hilbert spaces to HilbertC∗-modules. In this paper, we define and study the new concept of controlled continuous ∗-K-g-frames for Hilbert C∗-modules and we establish some properties.


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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 frame and Gabor frame [3, 4]. Frames have been used in signal processing, image processing, data compression, and sampling theory. The concept of a generalization of frame to a family indexed by some locally compact space endowed with a Radon measure was proposed by Kaiser [5] and independently by Ali et al. [6]. These frames are known as continuous frames. Gabardo and Han in [7] called them frames associated with measurable spaces, Askari-Hemmat et al. in [8] called them generalized frames, and in mathematical physics, they are known as coherent states.In 2012, Gavruta [9] introduced the notion of K-frame in Hilbert space to study the atomic systems with respect to a bounded linear operator K. Moreover, the following two recent papers [10, 11] are closely linked to Gravuta’s paper and represent a generalization of it by considering also unbounded operators.Controlled frames in Hilbert spaces have been introduced by Balazs et al. [12] to improve the numerical efficiency of iterative algorithms for inverting the frame operator.Frank and Larson [13] were the first one that extend frames to Hilbert C∗-modules.Controlled frames in HilbertC∗-modules was introduced by Kouchi and Rahimi [14], where the authors showed that they share many useful properties with their corresponding notions in a Hilbert space. Finally, we note that controlled K-g-frames in Hilbet spaces have been introduced by Hua and Huang [15]. The theory of continuous frames has been generalized in Hilbert C∗-modules. For more details, see [16–22].In this paper, we introduce the notion of a controlled continuous∗-K-g-frame for Hilbert C∗-modules.In the following, we briefly recall the definitions and basic properties ofC∗-algebras and Hilbert A-modules. Our references for C∗-algebras are [23, 24]. For a C∗-algebra A, if a∈A is positive, we write a≥0 and A+ denotes the set of all positive elements of A.Definition 1.
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 it 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∈HForx∈H, we definex=x,xA1/2. 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,xA1/2 for 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 2 [25].
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 the norm; i.e., there is m>0 such that ∥T∗x∥≥m∥x∥, x∈K(iii)
T∗ is bounded below with respect to the inner product; i.e., there is m′>0 such that(1)T∗x,T∗xA≥m′x,xA,x∈KLemma 3.
LetH and K be two Hilbert A-modules and T∈EndA∗H,K. Then, the following statements are equivalent:
(i)
The operatorT is bounded and A-linear(ii)
There exist0≤k such that Tx,TxA≤kx,xA for all x∈HLemma 4 [26].
LetH and K be two Hilbert A-modules and T∈End∗H,K. Then,
(i)
ifT is injective and T has a closed range, then the adjointable map T∗T is invertible and(2)∥T∗T−1∥−1≤T∗T≤∥T∥2(ii)
ifT is surjective, then the adjointable map TT∗ is invertible and(3)∥TT∗−1∥−1≤TT∗≤∥T∥2For the following theorem,RT denote the range of the operator T.Theorem 5 [27].
LetH be a Hilbert A-module over a C∗-algebra A and let T,S be two operators for EndA∗H. IfRS is closed, then the following statements are equivalent:
(4)RT⊂RS(i)
TT∗≤λ2SS∗ for some λ≥0(i)
There existsQ∈EndA∗H such that T=SQ


## 2. Controlled Continuous∗-K-g-Frames for Hilbert C∗-Modules

LetX be a Banach space, Ω,μ be a measure space, and f:Ω→X be a measurable function. Integral of the 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. Since every C∗-algebra and Hilbert C∗-module are a Banach space, thus we can use this integral and its properties.LetH and K be two Hilbert C∗-modules, Kw:w∈Ω is a family of subspaces of K, and EndA∗H,Kw is the collection of all adjointable A-linear maps from H into Kw. We define
(5)l2Ω,Kwω∈Ω=x=xww∈Ω:xw∈Kw,∫Ωxw2dμw<∞.For anyx=xw:w∈Ω and y=yw:w∈Ω, if the A-valued inner product is defined by x,y=∫Ωxw,ywAdμw, the norm is defined by ∥x∥=∥x,xA∥1/2. The l2Ω,Kwω∈Ω is a Hilbert C∗-module.LetA be a C∗-algebra; l2A is defined by
(6)l2A=aωw∈Ω⊆A:∥∫Ωaωaω∗dμω∥<∞.l2A is a Hilbert C∗-module (Hilbert A-module) with pointwise operations and the inner product defined by
(7)aωw∈Ω,bωw∈Ω=∫Ωaωbω∗dμω,aωw∈Ω,bωw∈Ω∈l2A,∥aωw∈Ω∥=∫Ωaωaω∗dμω1/2.LetGL+H be the set of all positive bounded linear invertible operators on H with bounded inverse.Definition 6 [16].
LetΛ=Λww∈Ω be a family in EndA∗H,Kw for all ω∈Ω, and C,C′∈GL+H. We say that the family Λis aC,C′-controlled continuous g-frame for Hilbert C∗-module H with respect to Kw:w∈Ω if it is a continuous g-Bessel family and there is a pair of constants 0<A,B such that, for any f∈H,
(8)Af,fA≤∫ΩΛwCf,ΛwC′fAdμw≤Bf,fA.A and B are called the C,C′-controlled continuous g-frame bounds.IfC′=I, then we call Λ=Λww∈Ω a C-controlled continuous g-frame for H with respect to Kw:w∈Ω.Definition 7.
LetC,C′∈GL+H. The family Λw∈EndA∗H,Kw:w∈Ω is called a C,C′-controlled continuous ∗-g-frame for Hilbert C∗-module H with respect to Vw:w∈Ω if Λ is continuous ∗-g-Bessel sequence and if there exist two strictly nonzero elements A and B in A such that
(9)Ax,xAA∗≤∫ΩΛwCx,ΛwC′xAdμw≤Bx,xAB∗,∀x∈U.A and B are called the C,C′-controlled continuous ∗-g-frame bounds.IfC′=I, then we call Λ a C-controlled continuous ∗-g-frames for H with respect to Kw:w∈Ω.Definition 8.
LetH be a Hilbert A-module over a unital C∗-algebra, and C,C′∈GL+H. A family of adjointable operators Λωw∈Ω⊂EndA∗H,Kw is said to be a C,C′-controlled continuous ∗-K-g-frame for Hilbert C∗-module H with respect to Kw:w∈Ω if there exist two strictly nonzero elements A and B in A such that
(10)AK∗f,K∗fAA∗≤∫ΩΛωCf,ΛωC′fAdμw≤Bf,fAB∗,f∈H.The elementsA and B are called continuous C,C′-controlled ∗-K-g-frame bounds.If(11)AK∗f,K∗fAA∗=∫ΩΛωCf,ΛωC′fAdμw,f∈H,we call this continuous C,C′-controlled ∗-K-g-frame a continuous C,C′-controlled tight K-g-frame, and if A=1A it is called a continuous C,C′-controlled Parseval ∗-K-g-frame. If only the right-hand inequality of (10) is satisfied, we call a continuous C,C′-controlled Bessel ∗-g-frame with Bessel bound B.Example 1.
LetH=M=x000y0/x,y∈ℂ, and A≪a00b/a,b∈ℂ.It is known thatA is a ℂ∗-algebra and H is a Hilbert A-module.LetC and C′ be two operators, respectively, defined as follows:
(12)C:H→H,M→αM,C′:H→H,M→βM, where α and β are two real numbers strictly positive.It is clear thatC,C′∈Gl+H.Indeed, for eachM∈H, one has
(13)C−1M=α−1M,C′−1M=β−1M.LetΩ=0,1 endowed with the Lebesgue measure. It is clear that there is a measure space. Moreover, for ω∈Ω, we define the operator Λw:H→H by
(14)ΛwM=0000wx0.Λw is linear, bounded, and selfadjoint.In addition, forM∈H, we have
(15)∫ΩΛwCM,ΛwC′MAdμω=∫Ωαβ0000wy0000wy¯00dμω=∫Ωαβw2000y2w2dμω=αβ3000y2≤αβ310010000y0,0000y0Aαβ31001,which show that the family Λωω∈Ω is a continuous C,C′-controlled Bessel sequence for H with αβ/3 as bounded.LetK:H→H by
(16)KM=0000y0.K=K∗ because K is a projection. Furthermore, we have
(17)αβ31001K∗M,K∗MA1001≤∫ΩΛwCM,ΛwC′MAdμω≤αβ31001M,MA1001,for every M∈H.Then,Λωω∈Ω is a continuous C,C′-controlled K-g-frame for H.Remark 9.
EveryC,C′-controlled continuous ∗-g-frame for H is a C,C′-controlled continuous ∗-K-g-frame for H.
Indeed, ifΛw∈EndA∗H,Kw:w∈Ω is a C,C′-controlled continuous ∗-g-frame for Hilbert C∗-module H with respect to Kw:w∈Ω, then there exist two strictly nonzero elements A and B in A such that
(18)Af,fAA∗≤∫ΩΛωCf,ΛωC′fAdμw≤Bf,fAB∗,f∈H.But(19)K∗f,K∗fA≤∥K∥2f,fA,f∈H.So,(20)A∥K∥−2K∗f,K∗fAA∗≤∫ΩΛωCf,ΛωC′fAdμw≤Bf,fAB∗,f∈H.Then,(21)A∥K∥−1K∗f,K∗fAA∥K∥−1∗≤∫ΩΛωCf,ΛωC′fAdμw≤Bf,fAB∗,f∈H.Hence,Λωw∈Ω is a controlled continuous ∗-K-g-frame for Hilbert C∗-module H with respect to Kw:w∈Ω with bounds A∥K∥−1 and B.Remark 10.
LetK∈EndA∗H be a surjective operator. Every C,C′-controlled continuous ∗-K-g-frame for H is a C,C′-controlled continuous ∗-g-frame for H.Proof.
LetK∈EndA∗H be a surjective operator; then, there exists m>0 such that
(22)mf,fA≤K∗f,K∗fA.
Hence,(23)Amf,fAAm∗≤AK∗f,K∗fAA∗.
SinceΛωw∈Ω is a C,C′-controlled continuous ∗-K-g-frame, then there exist two strictly nonzero elements A and B in A such that
(24)AK∗f,K∗fAA∗≤∫ΩΛωCf,ΛωC′fAdμw≤Bf,fAB∗,f∈H.
Therefore,(25)Amf,fAAm∗≤∫ΩΛωCf,ΛωC′fAdμw≤Bf,fAB∗,f∈H.
This gives thatΛωw∈Ω is a C,C′-controlled continuous ∗-g-frame with bounds Am and B, respectively.Proposition 11.
LetK∈EndA∗H and C,C′∈GL+H. Suppose that C and C′ commute with each other and commute with the operators Λω∗Λω for each ω∈Ω. A family Λωw∈Ω is a C,C′-controlled continuous ∗-K-g-frame Bessel family for H with respect to Kw:w∈Ω with bounds B if and only if the operator
(26)TC,C′yww∈Ω=∫ΩCC′1/2Λω∗yωdμw,yww∈Ω∈l2Ω,Kww∈Ωis well defined and bounded with ∥TC,C′∥≤∥B∥.Proof.
Assume thatΛωw∈Ω is a C,C′-controlled continuous Bessel ∗-K-g-frame family for H with respect to Kw:w∈Ω with bounds B. As a result of (2.3),
(27)∥∫ΩΛωCx,ΛωC′xA∥≤∥B∥2∥x,xA∥,∥TC,C′yωw∈Ω∥2=supx∈H,∥x∥=1∥∫ΩCC′1/2Λω∗yω,xAdμω∥2=supx∈H,∥x∥=1∥∫ΩCC′1/2Λω∗yω,xAdμω∥2=supx∈H,∥x∥=1∥∫Ωyω,ΛωCC′1/2xAdμω∥2≤supx∈H,∥x∥=1∥∫Ωyω,yωA∥∥∫ΩΛωCC′1/2x,ΛωCC′1/2xAdμω∥=supx∈H,∥x∥=1∥∫Ωyω,yωAdμω∥∥∫ΩΛωCx,ΛωC′xAdμω∥≤supx∈H,∥x∥=1∥∫Ωyω,yωAdμω∥∥B∥2∥x∥2=∥B∥2∥yωw∈Ω∥2.
Then, the sum∫ΩCC′1/2Λω∗yω is convergent and we have
(28)TC,C′yωw∈Ω2≤∥B∥2∥yww∈Ω∥2.
Hence,(29)∥TC,C′∥2≤∥B∥2.
Thus, the operatorTC,C′ is well defined and bounded and
(30)∥TC,C′∥≤∥B∥.
For the converse, suppose that the operatorTC,C′ is well defined and bounded and ∥TC,C′∥≤∥B∥. For all x∈H, we have
(31)∫ΩΛωCx,ΛωC′xAdμω=∫ΩC′Λω∗ΛωCx,xAdμω=∫ΩCC′1/2Λω∗ΛωCC′1/2x,xAdμω=TC,C′yωw∈Ω,xA≤∥TC,C′∥∥yωw∈Ω∥∥x∥≤∥TC,C′∥∫Ω∥ΛωCC′1/2x∥2dμω1/2∥x∥=∥TC,C′∥∫ΩΛωCx,ΛωC′xAdμω1/2∥x∥,where yω=ΛωCC′1/2x.
Therefore,(32)∫ΩΛωCx,ΛωC′xAdμω≤∥TC,C′∥2∥x∥2.
Hence,(33)∫ΩΛωCx,ΛωC′xA≤∥B∥2∥x∥2,as∥TC,C′∥≤∥B∥.
This gives thatΛωω∈Ω is a C,C′-controlled continuous ∗-K-g-frame Bessel family for H with respect to Kw:w∈Ω.LetΛ=Λωw∈Ω be a C,C′-controlled continuous Bessel ∗-K-g-frame for Hilbert C∗-module H over A with respect to Kw:w∈Ω with bounds A and B.We define the operatorTC,C′ by
(34)TC,C′:l2Ω,Kww∈Ω→H,such that
(35)TC,C′yww∈Ω=∫ΩCC′1/2Λω∗yωdμw,yww∈Ω∈l2Ω,Kww∈Ω.The bounded linear operatorTC,C′ is called the synthesis operator of Λ.The operator(36)TC,C′∗:H→l2Ω,Kww∈Ω,given by
(37)TC,C′∗x=ΛωC′C1/2xω∈Ω,x∈H,is called the analysis operator for Λ.Indeed, we have for allx∈H and yww∈Ω∈l2Ω,Kww∈Ω(38)TC,C′yww∈Ω,xA=∫ΩCC′1/2Λω∗yωdμw,xA=∫ΩCC′1/2Λω∗yω,xAdμw=∫Ωyω,ΛωCC′1/2xAdμw=yww∈Ω,ΛωC′C1/2xω∈Ωl2Ω,Kww∈Ω=yww∈Ω,TC,C′∗xl2Ω,Kww∈Ω,which shows that TC,C′∗ is the adjoint of TC,C′. If C and C′ commute between them and commute with the operators Λω∗Λω for each ω∈Ω, we define the C,C′-controlled continuous ∗-k-g-frame operator by
(39)SC,C′:H→H,x→SC,C′x=TC,C′TC,C′∗x=∫ΩC′Λw∗ΛwCxdμw.As consequence, one has the following proposition.Proposition 12.
The operatorSC,C′ is positive, selfadjoint, and bounded and
(40)∥A−1∥−2≤∥S∥≤∥B∥2.IfK is surjective, then SC,C′ is invertible.Proof.
By definition, we have∀x,y∈U(41)SC,C′x,y=∫ΩC′Λw∗ΛwCxdμw,y=∫ΩC′Λw∗ΛwCx,ydμw=∫Ωx,CΛw∗ΛwC′ydμw=x,∫ΩCΛw∗ΛwC′ydμw=x,SC′,Cy.
Then,S is a selfadjoint.
Clearly,S is positive.
By definition of aC,C′-controlled continuous ∗-K-g-frame, we have
(42)AK∗x,K∗xA∗≤∫ΩΛwCx,ΛwC′xdμw≤Bx,xB∗.
So(43)AK∗x,K∗xA∗≤SCC′x,x≤Bx,xB∗.
This gives(44)∥A−1∥−2∥KK∗x,x∥≤∥SCC′x,x∥≤∥B∥2∥x,x∥.
If we take supremum on allx∈H, where ∥x∥≤1, we have
(45)∥A−1∥−2∥K∥2≤∥S∥≤∥B∥2.
IfK is surjective, then, using Lemma 4, we have SC,C′ which is invertible.Lemma 13.
LetΛωw∈Ω⊂EndA∗H,Kw be a C,C′-controlled continuous Bessel ∗-K-g-frame family for Hilbert C∗-module H with respect to Kw:w∈Ω. Then, for any K∈EndA∗H such that KC=CK and KC′=C′K, the family ΛωKw∈Ω is a C,C′-controlled continuous Bessel ∗-K-g-frame for Hilbert C∗-module H.Proof.
Assume thatΛωw∈Ω is a C,C′-controlled continuous Bessel ∗-K-g-frame family for Hilbert C∗-module H with respect to Kw:w∈Ω with bound B. Then,
(46)∫ΩΛωCf,ΛωC′fAdμw≤Bf,fAB∗,f∈H.
So,(47)∫ΩΛωCKf,ΛωC′KfAdμw≤BKf,KfAB∗,f∈H.
Hence,(48)∫ΩΛωKCf,ΛωKC′fAdμw≤BKf,KfAB∗≤B∥K∥f,fAB∥K∥∗,f∈H.
The results hold.Lemma 14.
LetK∈EndA∗H and C,C′∈GL+H. Let Λωw∈Ω be a C,C′-controlled continuous Bessel ∗-K-g-frame for Hilbert C∗-module H with respect to Kw:w∈Ω. Λωw∈Ωis aC,C′-controlled continuous ∗-K-g-frame if and only if there exists A∈A such that
(49)AKK∗≤SC,C′.Proof.
The familyΛωw∈Ω is a C,C′-controlled continuous ∗-K-g-frame if and only if
(50)AK∗f,K∗fAA∗≤∫ΩΛωCf,ΛωC′fAdμw≤Bf,fAB∗,f∈H,if and only if
(51)AK∗f,AK∗fA≤SC,C′f,fA≤Bf,BfA,f∈H,if
(52)AK∗f,K∗fA≤Sf,fA,and the family Λωw∈Ω is a continuous C,C′-controlled Bessel K-g-frame sequence; then,
(53)Sf,fA≤Bf,fA,f∈H,which completes the proof.Theorem 15.
LetK∈EndA∗H and C,C′∈GL+H. Suppose that K∗ commute with C and C′. If Λωw∈Ω is a C,C′-controlled continuous ∗-g-frame for Hilbert C∗-module H with respect to Kw:w∈Ω, then ΛωK∗w∈Ω is a C,C′-controlled continuous ∗-K-g-frame for Hilbert C∗-module H with respect to Kw:w∈Ω with frame operator KSC,C′K∗.Proof.
LetΛωw∈Ω be a C,C′-controlled continuous ∗-g-frame for Hilbert C∗-module H with respect to Kw:w∈Ω; then,
(54)Af,fAA∗≤∫ΩΛωCf,ΛωC′fAdμw≤Bf,fAB∗,f∈H.
Hence,(55)AK∗f,K∗fAA∗≤∫ΩΛωCK∗f,ΛωC′K∗fAdμw≤BK∗f,K∗fAB∗,f∈H.
Therefore,(56)AK∗f,K∗fAA∗≤∫ΩΛωK∗Cf,ΛωK∗C′fAdμw≤B∥K∗∥2f,fAB∗,f∈H.
Thus,(57)AK∗f,K∗fAA∗≤∫ΩΛωK∗Cf,ΛωK∗C′fAdμw≤B∥K∗∥f,fAB∥K∗∥∗,f∈H.
This concludes thatΛωK∗w∈Ω is a C,C′-controlled continuous ∗-K-g-frame for Hilbert C∗-module H with respect to Kw:w∈Ω.
(58)KSC,C′K∗f=K∫ΩC′Λw∗ΛwCK∗fdμw=∫ΩKC′Λw∗ΛwCK∗fdμw=∫ΩC′KΛw∗ΛwK∗Cfdμw=∫ΩC′ΛwK∗∗ΛwK∗Cfdμw.Proposition 16.
LetK∈EndA∗H and C,C′∈GL+H. Suppose that C and C′ commute with each other and commute with S. Then, Λωw∈Ω is a C,C′-controlled continuous ∗-K-g-frame for H with respect to Kw:w∈Ω if and only if Λωw∈Ω is a C′C,IH-controlled continuous ∗-K-g-frame for Hilbert C∗-module H with respect to Kw:w∈Ω.Proof.
For allf∈H, we have
(59)C′−1SC,C′C−1f,fA=∫ΩC′Λω∗ΛωCC−1f,C′−1fAdμw=∫ΩΛω∗Λωf,fAdμw=Sf,fA,where
(60)Sf=∫ΩΛω∗Λωfdμw.
Hence,(61)S=C′−1SC,C′C−1.
For anyf∈H, we have
(62)∫ΩΛωCf,ΛωC′fAdμw=∫ΩC′Λω∗ΛωCf,fAdμw=SC,C′f,fA=C′SCf,fA=CSC′f,fA=SC′Cf,fA=∫ΩΛω∗ΛωC′Cf,fAdμw=∫ΩΛωC′Cf,ΛωfAdμw.
Hence,Λωw∈Ω is a continuous CC′,IH-controlled ∗-K-g-frame for H with bounds A and B with respect to Kw:w∈Ω if and only if
(63)AK∗f,K∗fAA∗≤∫ΩΛωC′Cf,ΛωfAdμw≤Bf,fAB∗,f∈H.
The results hold.Proposition 17.
LetK∈EndA∗H and C,C′∈GL+H. Then, Λωw∈Ω is a C,C′-controlled continuous ∗-K-g-frame for Hilbert C∗-module H with respect to Kw:w∈Ω if and only if Λωw∈Ω is a continuous C′C1/2,C′C1/2-controlled ∗-K-g-frame for H with respect to Kw:w∈Ω.Proof.
The proof is similar as the proof of Lemma 2.12.Proposition 18.
LetK∈EndA∗H and C,C′∈GL+H. Suppose that CK=KC, C′K=KC′, and CS=SC. Then, Λωw∈Ω is a C,C′-controlled continuous ∗-K-g-frame for H with respect to Kw:w∈Ω if and only if Λωw∈Ω is a continuous ∗-K-g-frame for H with respect to Kw:w∈Ω.Proof.
Assume thatΛωw∈Ω is a continuous ∗-K-g-frame for H with bounds A and B with respect to Kw:w∈Ω. Then,
(64)AKK∗f,fAA∗≤Sf,fA≤Bf,fAB∗,f∈H.
SinceC,C′ are bounded positive operators, there exist constants m,m′, M,M′0<m,m′,M,M′<∞ such that
(65)mIH≤C≤MIH,m′IH≤C′≤M′IH.
From(66)CSf,fA=f,SCfA=f,CSfA,we have
(67)mAKK∗≤SC=CS≤MBIH.
Then,(68)mm′AKK∗≤C′SC≤MM′BIH.
Therefore,(69)mm′AK∗f,K∗fAA∗≤∫ΩΛωCf,ΛωC′fAdμw≤MM′Bf,fAB∗,f∈H.
So(70)mm′AK∗f,K∗fAmm′A∗≤∫ΩΛωCf,ΛωC′fAdμw≤MM′Bf,fAMM′B∗,f∈H.
Hence,Λωw∈Ω is a continuous C,C′-controlled ∗-K-g-frame for H with respect to Kw:w∈Ω.
Conversely, assume thatΛωw∈Ω is a C,C′-controlled continuous ∗-K-g-frame for H with respect to Kw:w∈Ω with bounds A and B. On the one hand, we have for any f∈H,
(71)AK∗f,K∗fAA∗=ACC′1/2CC′−1/2K∗f,CC′1/2CC′−1/2K∗fAA∗=ACC′1/2K∗CC′−1/2f,CC′1/2K∗CC′−1/2fAA∗≤∥CC′1/2∥2∫ΩΛωCCC′−1/2f,ΛωC′CC′−1/2fAdμw=∥CC′1/2∥2SCCC′−1/2f,C′CC′−1/2fA=∥CC′1/2∥2SC1/2C′−1/2f,C′1/2C−1/2fA=∥CC′1/2∥2C−1/2C′1/2SC1/2C′−1/2f,fA=∥CC′1/2∥2Sf,fA.
So,(72)A∥CC′1/2∥−2K∗f,K∗fAA∗≤∫ΩΛωf,ΛωfAdμw=Sf,fA.
On the other hand, we have(73)∫ΩΛωf,ΛωfAdμw=Sf,fA=CC′−1/2CC′1/2Sf,fA=CC′1/2Sf,CC′−1/2fA=CC′CC′−1/2Sf,CC′−1/2fA=C′SCCC′−1/2f,CC′−1/2fA≤BCC′−1/2f,CC′−1/2fAB∗≤B∥CC′−1/2∥2f,fAB∗≤B∥CC′−1/2∥f,fAB∥CC′−1/2∥∗.
Therefore,Λωw∈Ω is a continuous ∗-K-g-frame for H with respect to Kw:w∈Ω with bounds A∥CC′1/2∥−2 and B∥CC′−1/2∥2.


## 3. Operators Preserving Controlled∗-K-g-Frames

In this section, for theC,C′-controlled ∗-K-g-frame Λωw∈Ω, we consider some proper relations between the operators T,K∈EndA∗H and C,C′∈GL+H and investigate the cases that ΛωTw∈Ω, ΛωT∗w∈Ω can also be C,C′-controlled ∗-K-g-frame. Next, by putting connections between the operators SC,C′, K, C, and C′, we reach to necessary and sufficient conditions that Λωw∈Ω can be a Parseval C,C′-controlled ∗-K-g-frame.Proposition 19.
LetK∈EndA∗H and C,C′∈GL+H. Let Λωw∈Ω be a C,C′-controlled continuous ∗-K-g-frame for H with respect to Kw:w∈Ω. Suppose that RK is closed. If T∈EndA∗H with RT⊂RK, then Λωw∈Ω is a C,C′-controlled continuous ∗-T-g-frame for H with respect to Kw:w∈Ω.Proof.
LetΛωw∈Ω be a C,C′-controlled continuous ∗-K-g-frame for H with respect to Kw:w∈Ω. Then, there exist A,B>0 such that
(74)AK∗f,K∗fAA∗≤∫ΩΛωCf,ΛωC′fAdμw≤Bf,fAA∗.
From Lemma 1.5 andRT⊂RK, there exists some m>0 such that
(75)TT∗≤mKK∗.
Hence,(76)AmT∗f,T∗fAA∗≤AK∗f,K∗fAA∗≤∫ΩΛωCf,ΛωC′fAdμw≤Bf,fAB∗.
So,Λωw∈Ω is a C,C′-controlled continuous ∗-T-g-frame for H with respect to Kw:w∈Ω.Theorem 20.
LetK∈EndA∗H be a surjective operator and Λw∈EndA∗H,Kw:w∈Ω be a C,C′-controlled continuous ∗-K-g-frame for H, with lower and upper bounds A and B, respectively, and with the C,C′-controlled continuous ∗-K-g-frame operator SC,C′. Let T∈EndA∗H be an invertible such that T commute with C and C′, then ΛwT:w∈Ω is a continuous ∗-K-g-frame for H with C,C′-controlled continuous ∗-K-g-frame operator T∗SC,C′T.Proof.
LetΛwω∈Ω be a C,C′-controlled continuous ∗-K-g-frame for H, with lower and upper bounds A and B, then
(77)AK∗Tx,K∗TxA∗≤∫ΩΛwCTx,ΛwC′Txdμw≤BTx,TxB∗,∀x∈H.
Using Lemma4, we have ∥T∗T−1∥−1x,x≤Tx,Tx, ∀x∈H.
K is surjective; then, there exist m such that
(78)mTx,Tx≤K∗Tx,K∗Tx,then
(79)m∥T∗T−1∥−1x,x≤K∗Tx,K∗Tx,so
(80)m∥T∗T−1∥−1Ax,xA∗≤AK∗Tx,K∗TxA∗,or ∥T−1∥−2≤∥T∗T−1∥−1; this implies
(81)∥T−1∥−1mAx,x∥T−1∥−1mA∗≤AK∗Tx,K∗TxA∗,∀x∈H.
And we know thatTx,Tx≤∥T∥2x,x, ∀x∈H. This implies that
(82)BTx,TxB∗≤∥T∥Bx,x∥T∥B∗,∀x∈H.
Using (77), (81), and (82), we have
(83)∥T−1∥−1mAx,x∥T−1∥−1mA∗≤∫ΩΛwCTx,ΛwC′Txdμw≤∥T∥Bx,x∥T∥B∗.
SoΛwT:w∈Ω is a C,C′-controlled continuous ∗-K-g-frame for H.
Moreover, for everyx∈H, we have
(84)T∗SC,C′Tx=T∗∫ΩC′Λw∗ΛwCTxdμw=∫ΩT∗C′Λw∗ΛwCTxdμw=∫ΩC′T∗Λw∗ΛwTCxdμw=∫ΩC′ΛwT∗ΛwTCxdμw.
This completes the proof.Corollary 21.
LetΛw∈EndA∗H,Kw:w∈Ω be a C,C′-controlled continuous ∗-K-g-frame for H and K∈EndA∗H be surjective, with continuous ∗-K-g-frame operator S. Then, ΛwS−1:w∈Ω is a C,C′-controlled continuous ∗-K-g-frame for H.Proof.
Result from the last theorem by takingT=SC,C′−1.Theorem 22.
LetK1,K2∈EndA∗H such that RK1⊥RK2. If Λωw∈Ω is a continuous C,C′-controlled ∗-K1-g-frame for Has well as∗-K2-g-frame for H with respect to Kw:w∈Ω and α and β are scalers. Then, Λωw∈Ω is a continuous C,C′-controlled ∗-αK1+βK2-g-frame and a continuous C,C′-controlled ∗-K1K2-g-frame for H with respect to Kw:w∈Ω.Proof.
LetΛωw∈Ω⊂EndA∗H,Kw be a controlled continuous ∗-K1-g-frame for H as well a ∗-K2-g-frame for H with respect to Kw:w∈Ω.
Then, there exist positive constantsA1,A2,B1,B2 such that
(85)A1K1∗f,K1∗fAA1∗≤∫ΩΛωCf,ΛωC′fAdμw≤B1f,fAB1∗,A2K2∗f,K2∗fAA2∗≤∫ΩΛωCf,ΛωC′fAdμw≤B2f,fAB2∗.
For anyf∈H, we have
αK1+βK2∗f,αK1+βK2∗fA =α¯K1∗f+β¯K2∗f,α¯K1∗f+β¯K2∗fA=α2K1∗f,K1∗fA+α¯βK1∗f,K2∗f+αβ¯K2∗f,K1∗f+β2K2∗f,K2∗f.
SinceRK1⊥RK2, then
(86)αK1+βK2∗f,αK1+βK2∗fA=α2K1∗f,K1∗f+β2K2∗f,K2∗fA.
LetB=α2A2A2∗+β2A1A1∗.
Therefore, for eachf∈H, we have
(87)A1A2B−1αK1+βK2∗f,αK1+βK2∗fAA1A2∗=A1A2B−1/2αK1+βK2∗f,αK1+βK2∗fAA1A2B−1/2∗≤∫ΩΛωCf,ΛωC′fAdμw≤B1+B2f,fAB1+B2∗.
Therefore,Λωw∈Ω is a continuous C,C′-controlled ∗-αK1+βK2-g-frame for H with respect to Kw:w∈Ω.
Also, for everyf∈H, we have
(88)K1K2∗f,K1K2∗fA=K2∗K1∗f,K2∗K1∗fA≤∥K2∗∥2K1∗f,K1∗fA.
SinceΛωw∈Ω is a continuous C,C′-controlled ∗-K1-g-frame for H with respect to Kw:w∈Ω, we have for every f∈H(89)A1∥K2∗∥−2K1K2∗f,K1K2∗fAA1∗≤∫ΩΛωCf,ΛωC′fAdμw≤B1f,fAB1∗.
Thus,(90)A1∥K2∗∥−1K1K2∗f,K1K2∗fAA1∥K2∗∥−1∗≤∫ΩΛωCf,ΛωC′fAdμw≤B1f,fAB1∗.
So,Λωw∈Ω is a continuous C,C′-controlled ∗-K1K2-g-frame for H with respect to Kw:w∈Ω.Corollary 23.
LetK∈EndA∗H. IfΛωw∈Ω is a continuous C,C′-controlled ∗-K-g-frame for H with respect to Kw:w∈Ω; then, for any operator ⊖ in the subalgebra generated by K, the family Λωw∈Ω is a continuous C,C′-controlled ∗-⊖-g-frame for H with respect to Kw:w∈Ω.



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*Source: 5548935-2021-04-29.xml*