# K-frames for Banach Spaces

Authors: Baoguang Sun; Chunyan Li
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In order to reconstruct elements from the range of a linear bounded operatorK on a separable Banach space, concepts of K-frames and K∗-atomic systems for Banach spaces are introduced, the relationship between the two is discussed, and the sufficient condition for a Banach space to have a K-frame is given. In addition, the duality of K-frames for Banach spaces is studied, the pair of K,K∗-dual Bessel sequences and the pair of approximate K,K∗-dual Bessel sequences for Banach spaces are introduced, and their important properties are discussed.


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## Body

## 1. Introduction

In 1952, Duffin and Schaeffer [1] introduced the notion of frames in nonharmonic Fourier analysis. Regrettably, their work was not continued until 1986 when Daubechies et al. [2] applied the theory of frames to wavelet and Gabor transform. Subsequently, the theory of frames began to be deeply studied and widely applied to signal processing, image processing, data compressing, sampling theory, and so on.In Hilbert spaces, some different concepts of frames, such as ordinary frames, frames of subspaces [3], and G-frames [4], have been proposed one after another in the recent 20 years. In 2012, Găvruţa [5] introduced the concept of K-frames for Hilbert spaces, which generalized the concept of ordinary frames for Hilbert spaces, to reconstruct elements from the range of a linear bounded operator K on a Hilbert space. In [6], Xiao et al. gave some methods to construct K-frames and discussed the stability of K-frames for Hilbert spaces. In [7], Neyshaburi and Arefijamaal focused on the reconstruction formulation to characterize all K-duals of a given K-frame and so gave several approaches for constructing K-frames for Hilbert spaces. In 2019, Jia and Zhu [8] studied the stabilities of K-frames under the operator perturbation. Furthermore, in [9], Shamsabadi and Arefijamaal studied some properties of K-frames, introduced the K-frame multipliers, and focused on representing elements from the range of K by K-frame multipliers. For more works about K-frames, please refer to Refs. [10–14].The theory of frames for Banach spaces originated from the works of Grochening [15]. In [15], Grochening introduced the notion of Banach frames and discussed the atomic decompositions versus frames. Subsequently, Aldroubi et al. [16] proposed p-frames for Banach spaces. In 2005, Casazza et al. [17] generalized the concept of p-frames and introduced Xd-frames for Banach spaces, where Xd was a BK-space. In 2008, Cao et al. [18] introduced the p,Y-operator frames for Banach spaces, where Y was a Banach space. p,Y-operator frames cover p-frames and Xd-frames. More works about the theory of frames for Banach spaces can be seen in Refs. [19–21]Inspired by the works in [5, 17] and in order to reconstruct elements from the range of a bound linear operator K on a separable Banach space, we will introduce the concepts of K-frames and K∗-atomic systems for Banach spaces. The concept of K-frames for Banach spaces is not only a generalization of Xd-frames but also a generalization of K-frames for Hilbert spaces. In this paper, we will discuss the relationship between K-frames and K∗-atomic systems for Banach spaces and study the sufficient condition for a Banach space to have a K-frame. In addition, in order to study the duality of K-frames for Banach spaces, we will introduce the concepts of the pair of K,K∗-dual Bessel sequences and the pair of approximate K,K∗-dual Bessel sequences for Banach spaces and discuss their important properties.The rest of the paper is organized as follows. In Section2, we review some existing concepts, such as BK-spaces, Xd-frames, and Xd∗-Riesz bases, and give several important lemmas, which will be used in proving the following main results. In Section 3, we introduce K-frames and K∗-atomic systems for separable Banach spaces, discuss the relationship between K-frames and K∗-atomic systems, and study some properties of K-frames. In Section 4, we introduce the pair of K,K∗-dual Bessel sequences and the pair of approximate K,K∗-dual Bessel sequences and study their properties.Throughout this paper, letF be the field ℂ of all complex numbers or the field ℝ of all real numbers, X,Y be separable Banach spaces, X∗ be the dual space of X, and X∗∗ be the dual of X∗. Assume that JX is the natural map of X into its second dual:
(1)JX:X⟶X∗∗,JXx=x^∈X∗∗,x∗,x^=x,x∗,∀x∗∈X∗,where x,x∗=x∗x and x∗,x^=x^x∗. Then, X can be seen as a subspace of X∗∗, and the map JX is a linear isometry with JXx=x. In this paper, we use LX,Y to denote the space of all bounded linear operators from X into Y. If X=Y, LX,Y is written as LX. For given T∈LX,Y, we denote the range of T by ranT.


## 2. Preliminaries

In this section, some important concepts and propositions that will be used in this paper are reviewed.A sequence spaceXd is called a BK-space (see [17]), if it is a Banach space and the coordinate functionals are continuous on Xd; i.e., the relations xn=αini=1∞,x=αii=1∞∈Xd,limn⟶∞xn=x imply limn⟶∞αin=αi,i=1,2,⋯.Example 1.
Letℂ be the set of all complex numbers. For 1≤p<∞, we denote
(2)ℓp=cii=1∞:ci∈ℂ,i=1,2,⋯,∑i=1∞cip<+∞,ℓ∞=cii=1∞:ci∈ℂ,i=1,2,⋯,supici<+∞.Then, all spacesℓp are BK-spaces.LetXd be a BK-space and the canonical unit vectors eii=1∞ form a Schauder basis for Xd; then, the following space
(3)Yd=ei,c∗i=1∞:c∗∈Xd∗with the norm ei,c∗i=1∞=c∗Xd∗ is also a BK-space. Moreover, Lemma 3.1 in [17] implies that Yd,· is isometrically isomorphic to Xd∗. In this paper, we always assume that the canonical unit vectors eii=1∞ form a Schauder basis for Xd and no longer distinguish between Yd and Xd∗.In [17], Casazza et al. introduced the concept of Xd-frames for Banach spaces. Let Xd be a BK-space. A family G=gii=1∞⊂X∗ is called an Xd-frame for X if it satisfies x,gii=1∞∈Xd,∀x∈X, and there exist constants A,B>0 such that
(4)Ax≤x,gii=1∞∥Xd≤Bx,∀x∈X,where A,B are, respectively, called a lower frame bound and an upper frame bound for G.Specifically, ifXd=ℓp,1<p<∞, we obtain the concept of p-frame for Banach spaces (see [16, 20]).In (4), if only the upper condition is satisfied, the family gii=1∞ is called an Xd-Bessel sequence for X. Let G=gii=1∞ be an Xd-Bessel sequence for X and the operator
(5)RG:X⟶Xdbe defined by
(6)RGx=x,gii=1∞,∀x∈X.It is clear thatRG is a linear bounded operator. RG is usually called the analysis operator of G. Furthermore, another important operator TG is defined as follows:
(7)TG:Xd∗⟶X∗,TGcii=1∞=∑i=1∞cigi,∀cii=1∞∈Xd∗.IfG is a Xd-Bessel sequence, it is easy to prove that TG is well defined and linear bounded. Here, TG is called the synthesis operator of G.A familygii=1∞⊂X∗ is said to be independent, if the following condition is satisfied:
(8)∑i=1∞cigi=0,cii=1∞∈Xd⇒ci=0,i=1,2,⋯.Definition 2.
LetX be a Banach space and Xd be a BK-space. A family G=gii=1∞⊂X∗ is called an Xd∗-Riesz basis for X∗ if the following conditions are satisfied:
(i)
span¯gi=X∗(ii)
There exist constantsC,D>0 such that(9)Ccii=1∞Xd∗≤∑i=1∞cigi≤Dcii=1∞Xd∗,∀cii=1∞∈Xd∗.
Specifically, whenXd=ℓp,1<p,q<∞,1/p+1/q=1, we call G a q-Riesz basis for X∗. Corollary 2.5 in [20] implied that a q-Riesz basis G with bounds C,D is always a p-frame for X and C,D are still its frame bounds.Lemma 3.
Assume thatT1∈LX1,X,T2∈LX2,X are two bounded operators and T2 is injective. If ranT1⊂ranT2, then there exists a bounded operator T∈LX1,X2 such that T1=T2T.Proof.
Suppose thatranT1⊂ranT2, then for every f∈X1, we have T1f∈ranT1⊂ranT2. Since T2 is injective, there exists a unique element h in X2 such that T2h=T1f. Set Tf=h. Then, T1=T2T. Now, we will prove that T is bounded.
In fact, letfi,hi be a sequence of elements in the graph of T such that limi⟶∞fi,hi=f,h. Since limi⟶∞T1fi=T1f and limi⟶∞T2hi=T2h, we have T1f=T2h. Thus, Tf=h. This shows that T is bounded on X1.Lemma 4 (von Neumann).
LetX be a Banach space, T∈LX, and I be the identity operator on X. If I−T<1, then T is invertible and T−1=I+∑k=1∞I−Tk.


## 3.K-frames and K∗-atomic Systems for Banach Spaces

In this section, we will introduceK-frames and K∗-atomic systems for Banach spaces and study the properties of K-frames and K∗-atomic systems.Definition 5.
LetXd be a BK-space and X be a Banach space, K∈LX. A family G=gii=1∞⊂X∗ is called a K-frame with respect to Xd for X if the following conditions are satisfied:
(i)
For eachx∈X, the sequence x,gii=1∞∈Xd(ii)
There exist constantsA,B>0 such that(10)AKx≤x,gii=1∞Xd≤Bx,∀x∈X,where A,B are, respectively, called a lower frame bound and an upper frame bound of the K-frame G. In particular, G is called a tight K-frame if the following condition is satisfied:
(11)AKx=x,gii=1∞Xd,∀x∈X.Example 6.
LetX=Xd=ℓ1. Then, the canonical unit vectors sequence eii=1∞⊂X∗=ℓ∞. For every x=cii=1∞∈X, we define
(12)Kx=Kcii=1∞=c2,c3⋯,cn+1,⋯.
Clearly,K is a bound linear operator. Thus, we have
(13)x,eii=1∞Xd=cii=1∞ℓ1=∑i=1∞ci=x,Kxℓ1=c2,c3,⋯,cn+1,⋯ℓ1=∑i=2∞ci≤∑i=1∞ci=x=x,eii=1∞Xd.
This shows thateii=1∞ is a K-frame with respect to Xd for X.Remark 7.
Since anXd-frame G=gii=1∞⊂X∗ for X must be an I-frame for X, where I is the identity operator on X, the concept of K-frames for Banach spaces is a generalization of the concept of Xd-frames.
Letεii=1∞ be a Schauder basis for X. We define the space as follows:
(14)Y=cii=1∞:ci∈F,i=1,2,⋯,∑i=1∞ciεiconvergesinX,cii=1∞Y=supN∑i=1Nciεi,and the operator
(15)S:Y⟶X,Scii=1∞=∑i=1∞ciεi,∀cii=1∞∈Y.
Then,Y,·Y is a Banach space and S is a topological isomorphic mapping. Moreover, we can prove that Y is a BK-space. Indeed, let Pn be the n-th coordinate functional on Y, i.e., Pncii=1∞=cn,n=1,2,⋯. Suppose that cii=1∞Y⟶0. Then, for every N, we have ∑i=1Nciεi⟶0. When N=1, we get c1ε1=c1·ε1⟶0ε1≠0, so c1⟶0. If N=2, we have ∑i=12ciεi⟶0. Since
(16)c2ε2=c1ε1+c2ε2−c1ε1≤∑i=12ciεi+c1ε1⟶0,we get c2⟶0. And so on, cn⟶0,∀n. This shows that every coordinate functional Pnn=1,2,⋯ is continuous on Y. Hence, Y is a BK-space.Theorem 8.
LetX be a separable Banach space with a Schauder basis and K∈LX. Then, X has a K-frame.Proof.
Letεii=1∞ be a Schauder basis for X. Then, for all x∈X, we have
(17)x=∑i=1∞aixεi=∑i=1∞x,aiεi,where ai are the coefficient functionals and aii=1∞,εii=1∞ are biorthogonal.
Putgi=K∗ai,i=1,2,⋯; then, for every x∈X, we have
(18)∑i=1∞gixεi=∑i=1∞x,giεi=∑i=1∞x,K∗aiεi=∑i=1∞Kx,aiεi=∑i=1∞aiKxεi=Kx.
This shows that∑i=1∞gixεi converges in X and gixi=1∞∈Y, where Y is the following space:
(19)Y=cii=1∞:ci∈F,i=1,2,⋯,∑i=1∞ciεiconvergesinX.
Define the operator(20)Scii=1∞=∑i=1∞ciεi.
Then, for everyx∈X, we have
(21)x,gii=1∞Y=supN∑i=1Nx,giεi=supN∑i=1Nx,K∗aiεi=supN∑i=1NKx,aiεi=supN∑i=1NaiKxεi=aiKxi=1∞Y=S−1Kx≤S−1·K·x.
Moreover, we get(22)Kx=∑i=1∞aiKxεi=∑i=1∞Kx,aiεi=∑i=1∞x,K∗aiεi=∑i=1∞x,giεi=∑i=1∞gixεi=limN⟶∞∑i=1Ngixεi≤supN∑i=1Ngixεi=gixi=1∞Y=x,gii=1∞Y.
The above two inequalities yield that(23)Kx≤gixi=1∞Y≤Bx,∀x∈X,where B=S−1·K. Hence, gii=1∞ is a K-frame with respect to Y for X.Definition 9.
LetX be a Banach space and K∈LX. A family gii=1∞⊂X∗ is called a K∗-atomic system with respect to Xd∗ for X∗ if the following conditions are satisfied:
(i)
For everycii=1∞∈Xd∗, ∑i=1∞cigi converges(ii)
For everyx∗∈X∗, there exist a sequence cx∗=cii=1∞∈Xd∗ and a constant C>0 such that cii=1∞≤Cx∗ and K∗x∗=∑i=1∞cigi
Now, we will discuss the relationship betweenK-frames and K∗-atomic systems for Banach spaces.Theorem 10.
LetXd be a BK-space and gii=1∞⊂X∗ be a K∗-atomic system with respect to Xd∗ for X∗; then, it is a K-frame with respect to Xd for X.Proof.
Suppose thatgii=1∞ is a K∗-atomic system with respect to Xd∗ for X∗. Then,
(24)Kx=supx∗≤1Kx,x∗=supx∗≤1x,K∗x∗.
From the definition of theK∗-atomic system, it follows that there exist a sequence cx∗=cii=1∞∈Xd∗ and a constant C>0 such that cii=1∞Xd∗≤Cx∗ and K∗x∗=∑i=1∞cigi. Hence, we have
(25)Kx=sup∥x∗∥≤1Kx,x∗=sup∥x∗∥≤1x,K∗x∗=sup∥x∗∥≤1x,∑i=1∞cigi=sup∥x∗∥≤1∑i=1∞cix,gi≤sup∥x∗∥≤1x,gii=1∞,cii=1∞=sup∥x∗∥≤1cii=1∞Xd∗·x,gii=1∞Xd≤sup∥x∗∥≤1Cx∗·x,gii=1∞Xd=Cx,gii=1∞Xd.
On the other hand, define the operator:(26)T:Xd∗⟶X∗,Tcii=1∞=∑i=1∞cigi.
Then,∑i=1∞cigi converges in X∗. By the Banach-Steinhaus theorem, we can conclude that T is bounded. Denote the adjoint of the operator T by T∗; then, for every cii=1∞∈Xd∗ and x∈X, the operator T∗ can be determined by the following calculation:
(27)x,Tcii=1∞=x,∑i=1∞cigi=∑i=1∞x,cigi=∑i=1∞cix,gi=x,gii=1∞,cii=1∞=T∗x,cii=1∞,i.e.,
(28)T∗x=x,gii=1∞∈Xd.
Thus, we have(29)x,gii=1∞Xd=T∗x≤T∗·x,∀x∈X.
Therefore,gii=1∞ is a K-frame with respect to Xd for X.Theorem 11.
Let1<p<∞ and G=gii=1∞ be a K-frame with respect to ℓp for X. If G is independent and the operators K,TG satisfy ranK∗⊂ranTG, then G is a K∗-atomic system with respect to ℓq for X∗, where TG is the synthesis operator of G.Proof.
Suppose thatgii=1∞ is a K-frame with respect to ℓp for X with frame bounds A,B>0 and TG is the synthesis operator of G; then, we have
(30)TG:ℓq⟶X∗,TGcii=1∞=∑i=1∞cigi.
Since the canonical unit vectorseii=1∞ form a Schauder basis for ℓq, we can get
(31)gi=TGei,i=1,2,⋯.
Assume thatG is independent and ranK∗⊂ranTG; then, the operator TG is injective and there exists a bounded operator C∈LX∗,ℓq such that K∗=TGC by using Lemma 3. For every x∗∈X∗, since Cx∗∈ℓq, we denote the i-th element of Cx∗ by ci,i=1,2,⋯; then,
(32)cii=1∞=Cx∗≤C·x∗,and we have
(33)K∗x∗=TGCx∗=∑i=1∞cigi,∀x∗∈X∗.
This shows thatgii=1∞ is a K∗-atomic system with respect to ℓq for X∗.
We give a sufficient condition for ap-Bessel sequence to become a K-frame by choosing a linear bounded operator K.Theorem 12.
Let1<p,q<∞,1/p+1/q=1 and G=gii=1∞⊂X∗ be a p-Bessel sequence for X. If X has a p-Riesz basis, then there exists a linear bounded operator K∈LX such that G is a K-frame with respect to ℓp for X.Proof.
LetF=fii=1∞⊂X be a p-Riesz basis for X with bounds A,B. By using Corollary 2.5 in [20], F is a q-frame for X∗ with bounds A,B. In addition, from Theorem 2.8 in [20], it follows that there exists a unique q-Riesz basis hii=1∞⊂X∗ for X∗ such that
(34)x=∑i=1∞x,hifi,∀x∈X,x∗=∑i=1∞fi,x∗hi,∀x∗∈X∗.
SetUhi=gi,i=1,2,⋯; then, U is a bounded operator from X∗ to X∗. Indeed, for every x∗∈X∗, we have
(35)U∑i=1Nfi,x∗hi=∑i=1Nfi,x∗Uhi=∑i=1Nfi,x∗gi.
SinceF is a q-frame for X∗, we have fi,x∗i=1∞∈ℓq. Hence, when N⟶∞, we can get
(36)U∑i=1∞fi,x∗hi=∑i=1∞fi,x∗gi=Ux∗.
This shows thatU is well defined. Let RF be the analysis operator of F and TH be the synthesis operator of H. The above equation yields that U=TGRF. Hence, U is bounded.
PutK=U∗; then, for every x∈X, we have
(37)Kx=supx∗≤1Kx,x∗=supx∗≤1∑i=1∞Kx,hifi,x∗=supx∗≤1∑i=1∞x,K∗hifi,x∗=supx∗≤1∑i=1∞x,Uhifi,x∗=supx∗≤1∑i=1∞x,gifi,x∗≤supx∗≤1∑i=1∞x,gip1/p·∑i=1∞fi,x∗q1/q≤B∑i=1∞x,gip1/p.
Sincegii=1∞⊂X∗ is a Bessel sequence, the above inequality yields that it is a K-frame with respect to ℓp for X.


## 4. The Duality ofK-frames

In this section, we will study the duality ofK-frames for Banach spaces and reconstruct elements from the range of a linear bounded operator K on a separable Banach space.Definition 13.
LetK∈LX, G=gii=1∞⊂X∗ be an Xd-Bessel sequence for X, and F=fii=1∞ be a Xd∗-Bessel sequence for X∗. F,G is called a pair of K,K∗-dual Bessel sequences if the following conditions are satisfied:
(38)Kx=∑i=1∞x,gifi,∀x∈X,(39)K∗x∗=∑i=1∞fi,x∗gi,∀x∗∈X∗.
In fact, ifF,G form a pair of K,K∗-dual Bessel sequences, they are a K∗-frame and a K-frame, respectively.Proposition 14.
LetK∈LX, G=gii=1∞⊂X∗ be an Xd-Bessel sequence, and F=fii=1∞⊂X be an Xd∗-Bessel sequence. If F,G is a pair of K,K∗-dual Bessel sequences, then G is a K-frame with respect to Xd for X and F is a K∗-frame with respect to Xd∗ for X∗.Proof.
Suppose thatF=fii=1∞ has a Bessel bound B1; then,
(40)fi,x∗i=1∞Xd∗≤B1x∗,∀x∗∈X∗.
SinceF,G is a pair of K,K∗-dual Bessel sequences, for all x∗∈X∗, we have
(41)K∗x∗=∑i=1∞fi,x∗gi.
So(42)K∗x∗=supx≤1x,K∗x∗=supx≤1x,∑i=1∞fi,x∗gi=supx≤1∑i=1∞fi,x∗x,gi=supx≤1fi,x∗i=1∞,x,gii=1∞≤supx≤1fi,x∗i=1∞Xd∗·x,gii=1∞Xd≤supx≤1B2·x·fi,x∗i=1∞Xd∗≤B2·fi,x∗i=1∞Xd∗,where TG is the analysis operator of the Bessel sequence G and B2 is its Bessel bound. Combining the above inequality and (40), we obtain
(43)B2−1K∗x∗≤fi,x∗i=1∞Xd∗≤B1x∗,∀x∗∈X∗.
Therefore,F is a K∗-frame with respect to Xd∗ for X∗.
Similarly, we can prove thatG is a K-frame with respect to Xd for X.
From the above proposition, when (38) and (39) are satisfied, F=fii=1∞ and G=gi∗i=1∞ are, respectively, a K∗-frame and a K-frame. Therefore, F,G is also called a pair of K,K∗-dual frames. Moreover, F is called a K∗-dual of G and G is called a K-dual of F.Theorem 15.
Let1<p<∞ and G=gi∗i=1∞⊂X∗ be a K-frame with respect to ℓp for X. If its analysis operator RG is invertible, then G must have a K∗-dual.Proof.
Since the analysis operatorRG is invertible, for every x∈X, we have RG−1RGx=x. Since the canonical unit vectors eii=1∞ form a Schauder basis for the BK-space ℓp, we can set fi=KRG−1ei,i=1,2,⋯; then,
(44)x∗,fii=1∞=x∗,KRG−1eii=1∞=RG−1∗K∗x∗,eii=1∞=RG−1∗K∗x∗=RG∗−1K∗x∗∈ℓq,∀x∗∈X∗.
So we obtainx∗,fii=1∞ℓq=RG∗−1K∗x∗≤RG∗−1·K∗·x∗. This shows that fii=1∞ is a q-Bessel sequence 1/p+1/q=1. In addition, for every x∈X, we have
(45)Kx=KRG−1RGx=KRG−1x,gii=1∞=KRG−1∑i=1∞x,giei=∑i=1∞x,giKRG−1ei=∑i=1∞x,gifi.i.e.,
(46)Kx=∑i=1∞x,gifi,∀x∈X.
Sincefi,x∗i=1∞∈ℓq∀x∗∈X∗, for any ε>0, there exists a positive integer N such that ∀n,m>N, ∑i=nmfi,x∗ei<ε/B, where B is the upper frame bound. Hence, we have
(47)∑i=nmfi,x∗gi=supx≤1x,∑i=nmfi,x∗gi=supx≤1∑i=nmfi,x∗x,gi≤supx≤1∑i=nmfi,x∗q1/q·∑i=nmx,gip1/p≤supx≤1∑i=nmfi,x∗q1/q·∑i=1∞x,gip1/p≤B·∑i=nmfi,x∗q1/q=B·∑i=nmfi,x∗ei<ε.
This shows that∑i=1∞fi,x∗gi converges in X∗. Here, we get
(48)x,K∗x∗=Kx,x∗=∑i=1∞x,gifi,x∗=∑i=1∞x,gifi,x∗=x,∑i=1∞fi,x∗gi.
Thus,(49)K∗x∗=∑i=1∞fi,x∗gi,∀x∗∈X∗.
From (46) and (49), it follows that fii=1∞ is the K∗-dual of G.
Although the dual of a frame for Hilbert spaces always exists, to find it is very complicated. In [22], Christensen and Laugesen introduced the concept of an approximate dual to solve such problem. For Banach spaces, the duality of all kinds of frames such as p-frames, Xd-frames, p,Y-operator frames, or K-frames, does not necessarily exist. Therefore, for a K-frame G=gii=1∞⊂X∗, we aim to identify a sequence fii=1∞⊂X, following the approximate series expansion of Kx and K∗x∗:
(50)Kx≈∑i=1∞x,gifi,∀x∈X,(51)K∗x∗≈∑i=1∞fi,x∗gi,∀x∗∈X∗.
To realize the approximate reconstruction of elements from the range of the linear bounded operator on a Banach space, the concept of approximateK,K∗-dual Bessel sequences for Banach spaces will be introduced.Definition 16.
LetK∈LX, G=gii=1∞⊂X∗ be an Xd-Bessel sequence for X, and F=fii=1∞ be an Xd∗-Bessel sequence for X∗. F,G is called a pair of approximate K,K∗-dual Bessel sequences if the following conditions are satisfied:
(52)Kx−∑i=1∞x,gifi≤εx,∀x∈X,(53)K∗x∗−∑i=1∞fi,x∗gi≤εx∗,∀x∗∈X∗,where ε is a small positive number.Remark 17.
To ensure that (50) and (51) are satisfied, the constant εin (52) and (53) is usually set to a very small value. Assume that ε<1; we may define the concept of approximate K,K∗-dual Bessel sequences by the analysis operator and the synthesis operator of Bessel sequences.Definition 18.
LetK∈LX, G=gii=1∞⊂X∗ be an Xd-Bessel sequence for X, and F=fii=1∞ be an Xd∗-Bessel sequence for X∗. F,G is called a pair of approximate K,K∗-dual Bessel sequences if K−TFRG<1 and K∗−TGRF<1.
IfK∈LX is injective, then for every y∈ranK, there exists a unique xy∈X such that Kxy=y. Thus, we can define an operator K as follows:
(54)K:ranK⊂X⟶X,Ky=xy,∀y∈ranK.
It is obvious thatK is invertible on ranK and satisfies the following equations:
(55)KKx=x,∀x∈X,KKy=y,∀y∈ranK.
Furthermore, if the operatorK∈LX has a generalized inverse K†, then it satisfies that
(56)KK†K=K,K†KK†=K†.
The concept of the generalized inverse of a linear bounded operator in Banach spaces can be seen Ref. [23].Lemma 19.
LetK∈LX,K>1, G=gii=1∞⊂X∗ be an Xd-Bessel sequence, F=fii=1∞⊂X be an Xd∗-Bessel sequence, and F,G be a pair of approximate K,K∗-dual Bessel sequences. If K is injective and has a generalized inverse K†, then the operator K†TFRG is invertible.Proof.
Suppose thatF,G is a pair of approximate K,K∗-dual Bessel sequences, then K−TFRG<1. Since K≥1 and it is injective and has a generalized inverse K†, for every x∈X, we have
(57)I−K†TFRG=supx≤1x−K†TFRGx=supx≤1KKx−KKK†TFRGx=supx≤1KKK†Kx−KKK†TFRGx≤KKK†·supx≤1Kx−TFRGx≤K†·K−TFRG<1.
From Lemma4, it follows that K†TFRG is invertible on X.Proposition 20.
LetK∈LX,K≥1, G=gii=1∞⊂X∗ be an Xd-Bessel sequence, F=fii=1∞⊂X be an Xd∗-Bessel sequence, and F,G be a pair of approximate K,K∗-dual Bessel sequences. If K is injective and has a generalized inverse K†, then G is a K-frame with respect to Xd for X.Proof.
SinceF,G is a pair of approximate K,K∗-dual Bessel sequences, from Lemma 19, it follows that K†TFRG is invertible. For every x∈X, we have
(58)x=K†TFRG−1K†TFRGx=K†TFRG−1K†TFRGx=K†TFRG−1K†∑i=1∞x,gifi=K†TFRG−1∑i=1∞x,giK†fi=∑i=1∞x,giK†TFRG−1K†fi.
Suppose that the Bessel bound ofF is B; then, we have
(59)Kx=supx∗≤1Kx,x∗=supx∗≤1x,K∗x∗=supx∗≤1∑i=1∞x,giK†TFRG−1K†fi,K∗x∗=supx∗≤1∑i=1∞x,giK†TFRG−1K†fi,K∗x∗=supx∗≤1∑i=1∞x,gifi,K†TFRG−1K†∗K∗x∗=supx∗≤1x,gii=1∞,fi,K†TFRG−1K†∗K∗x∗i=1∞≤supx∗≤1x,gii=1∞Xd·fi,K†TFRG−1K†∗K∗x∗i=1∞Xd∗≤supx∗≤1B·K†TFRG−1K†∗K∗x∗·x,gii=1∞Xd≤B·K†TFRG−1K†∗K∗·x,gii=1∞Xd≤β·x,giXd,where β=B·K†TFRG−1K†∗K∗≠0. Thus, we obtain
(60)1βKx≤x,giXd,∀x∈X,
SinceG is an Xd-Bessel sequence, it is a K-frame with respect to Xd for X.Remark 21.
Similarly, in the above proposition, under some conditions,F can become a K∗-frame with respect to Xd∗ for X∗.Theorem 22.
LetK∈LX,K≥1, G=gi∗i=1∞⊂X∗ be an Xd-Bessel sequence, F=fii=1∞⊂X be an Xd∗-Bessel sequence, and F,G be a pair of approximate K,K∗-dual Bessel sequences. If K is injective and has a generalized inverse K†, then
(i)
The sequenceKK†TFRG−1K†fii=1∞ is a K∗-dual of G, and the operator KK†KTFRG−1K† can be written as(61)KK†TFRG−1K†fi=KK†fi+∑k=1∞KI−K†TFRGkK†fi,i=1,2,⋯.(ii)
For every fixed natural numbern, we put(62)γin=KK†fi+∑k=1nKI−K†TFRGkK†fi,i=1,2,⋯.Then, Yn,G is a pair of approximate K,K∗-dual Bessel sequences, where Yn=γini=1∞.Proof.
(i)
Suppose thatF,G is a pair of approximate K,K∗-dual Bessel sequences; from Lemmas 4 and 19, it follows that K†TFRG is invertible and(63)K†TFRG−1=I+∑k=1∞I−K†TFRGk.
Thus, for everyfi,i=1,2,⋯, we have
(64)KK†TFRG−1K†fi=KK†fi+∑k=1∞KI−K†TFRGkK†fi.
From the proof of Proposition20, it follows that
(65)Kx=K∑i=1∞x,giK†TFRG−1K†fi=∑i=1∞x,giKK†TFRG−1K†fi,∀x∈X.
On the other hand, for everyx∈X,x∗∈X∗, we have
(66)x,K∗x∗=Kx,x∗=∑i=1∞x,giKK†TFRG−1K†fi,x∗=∑i=1∞x,gi·KK†TFRG−1K†fi,x∗=x,∑i=1∞KK†TFRG−1K†fi,x∗gi.
That is,(67)K∗x∗=∑i=1∞KK†TFRG−1K†fi,x∗gi,∀x∗∈X∗.
Therefore,KK†TFRG−1K†fii=1∞ is a K∗-dual of G.
(ii)
Given a fixed natural numbern, set(68)γin=KK†fi+∑k=1nKI−K†TFRGkK†fi,i=1,2,⋯.Then, for every x∗∈X∗, we have
(69)γin,x∗i=1∞Xd∗=KK†fi+∑k=1nKI−K†TFRGkK†fi,x∗i=1∞Xd∗=KK†fi,x∗+∑k=1nKI−K†TFRGkK†fi,x∗i=1∞Xd∗≤fi,KK†∗x∗i=1∞Xd∗+∑k=1nKI−K†TFRGkK†fi,x∗i=1∞Xd∗≤B·KK†∗x∗+fi,∑k=1nKI−K†TFRGkK†∗x∗i=1∞Xd∗≤B·x∗+B·∑k=1nKI−K†TFRGkK†∗x∗≤B·x∗+B·∑k=1nKI−K†TFRGkK†∗·x∗=B1+∑k=1nKI−K†TFRGkK†∗·x∗=M·x∗,where B is the Bessel bound of F and M=B1+∑k=1nKI−K†TFRGkK†∗. This shows that Yn is an Xd∗-Bessel sequence.
LetTYn be the synthesis operator of Yn. For every x∈X, we have
(70)TYnRGx=∑i=1∞x,giγin=∑i=1∞x,gi·∑k=0nKI−K†TFRGkK†fi=∑k=0nKI−K†TFRGkK†∑i=1∞x,gifi=∑k=0nKI−K†TFRGkK†TFRGx=∑k=0nKI−K†TFRGk·K†TFRGx=∑k=0nKI−K†TFRGk·I−I−K†TFRGx=∑k=0nKI−K†TFRGk−I−K†TFRGk+1x=Kx−KI−K†TFRGn+1x.That is,
(71)TYnRGx=Kx−KI−K†TFRGn+1x.
Thus,(72)Kx−TYnRGx=KI−K†TFRGn+1x.
It follows that(73)K−TYnRG=supx≤1Kx−TYnRGx=supx≤1KI−K†TFRGn+1x≤K·supx≤1I−K†TFRGn+1x=K·I−K†TFRGn+1≤K·I−K†TFRGn+1≤K·KK−KKK†TFRGn+1=K·KKK†K−KKK†TFRGn+1≤K·KKK†n+1·K−TFRGn+1≤K·K†n+1·K−TFRGn+1.
SinceK−TFRG<1 and K≥1, we obtain K−TYnRG<1. Similarly, we can prove that K∗−TGRYn<1. Hence, Yn,G is a pair of approximate K,K∗-dual Bessel sequences.
In fact, from part (ii) of Theorem22, we find that as n increases, Yn=γini=1∞ is closer to a K∗-dual of G. Hence, by choosing n sufficiently large, the reconstruction of Kx is “almost perfect.” If we denote γi=limn⟶∞γin,i=1,2,⋯, then γii=1∞ happens to be a K∗-dual of G.Theorem 23.
LetK∈LX, F=fii=1∞ be an Xd∗-Bessel sequence with the bound B, G=gii=1∞ be an Xd-Bessel sequence for X, and F,G be a pair of K,K∗-dual Bessel sequences. For a given sequence H=hii=1∞⊂X∗, if x,hii=1∞∈Xd and there exists a constant λ>0 such that λ<1/B and
(74)x,gi−hii=1∞Xd≤λx,∀x∈X,then F,H is a pair of approximate K,K∗-dual Bessel sequences.Proof.
LetTF be the synthesis operator of F and RG be the analysis operator of G. For every x∈X, we have
(75)x,hii=1∞Xd≤x,hi−gii=1∞Xd+x,gii=1∞Xd≤λx+B1x≤λ+B1x,where B1 is the upper frame bound of G. Thus, H is an Xd∗-Bessel sequence. Suppose that RH is the analysis operator of H. Then, (74) complies that
(76)RG−RH≤λ.
In addition, for allcii=1∞∈Xd∗, we have
(77)TG−THcii=1∞=supx≤1x,TG−THcii=1∞=supx≤1x,∑i=1∞cigi−hi=supx≤1∑i=1∞cix,gi−hi=supx≤1x,gi−hii=1∞,cii=1∞≤supx≤1x,gi−hii=1∞·cii=1∞=supx≤1RG−RHx·cii=1∞≤RG−RH·cii=1∞≤λ·cii=1∞.That is,
(78)TG−TH≤λ.
SinceF,G is a pair of K,K∗-dual Bessel sequences, for every x∈X,x∗∈X∗, we have
(79)Kx=∑i=1∞x,gifi,K∗x∗=∑i=1∞fi,x∗gi,i.e.,
(80)K=TFRG,K∗=TGRF.
Hence, we obtain(81)K−TFRH=TFRG−TFRH≤TF·RG−RH≤Bλ<1,K∗−THRF=TGRF−THRF≤TG−TH·RF≤Bλ<1.
It proves thatF,H is a pair of approximate K,K∗-dual Bessel sequences.Theorem 24.
LetK∈LX, F=fii=1∞ be an Xd∗-Bessel sequence with bound B1, G=gii=1∞ be an Xd-Bessel sequence for X with the bound B2, and H=hii=1∞⊂X∗ be an Xd-Bessel sequence. If F,G is a pair of K,K∗-dual Bessel sequences and there exist constants λ1,μ>0,1>λ2>0 such that
(82)B1B21+λ1+B1μ1−λ2<1−λ1K,(83)∑i=1ncigi−hi≤λ1∑i=1ncigi+λ2∑i=1ncihi+μcii=1∞,∀n∈N,∀cii=1∞∈Xd∗,then F,H is a pair of approximate K,K∗-dual Bessel sequences.Proof.
LetTF, TH, and TG be, respectively, the synthesis operators of F,H,G and RG be the analysis operator of G; then, we have TG≤B2 and RG≤B2. From (83), for every cii=1∞∈Xd∗, it follows that
(84)TGcii=1∞−THcii=1∞≤λ1TGcii=1∞+λ2THcii=1∞+μcii=1∞.
Thus, we have(85)THcii=1∞≤1+λ11−λ2TGcii=1∞+μ1−λ2cii=1∞≤1+λ11−λ2TG·cii=1∞+μ1−λ2cii=1∞≤1+λ11−λ2B2·cii=1∞+μ1−λ2cii=1∞≤1+λ1B2+μ1−λ2cii=1∞.
SoTH≤1+λ1B2+μ/1−λ2.
Next, we will prove thatF,H is a pair of approximate K,K∗-dual Bessel sequences. In fact, for every x∗∈X∗, take cii=1∞=RFx∗; by (83), we have
(86)TGRFx∗−THRFx∗≤λ1TGRFx∗+λ2THRFx∗+μRFx∗.
SinceF,G is a pair of K,K∗-dual Bessel sequences, for every x∈X,x∗∈X∗, we have
(87)Kx=∑i=1∞x,gifi,K∗x∗=∑i=1∞fi,x∗gi,i.e.,
(88)K=TFRG,K∗=TGRF.
Combining (82), (83), and (86), we have
(89)K∗x∗−THRFx∗≤λ1K∗x∗+λ2THRFx∗+μRFx∗≤λ1K∗·x∗+λ2TH·B1·x∗+μB1·x∗≤λ1K∗·x∗+λ21+λ1B2+μ1−λ2·B1·x∗+μB1·x∗=λ1K∗+λ21+λ1B2B1+λ2μB11−λ2+μB1x∗≤x∗.
Thus,K∗−THRF<1.
On the other hand, for everyx∈X, we have
(90)Kx−TFRHx=supx∗≤1Kx−TFRHx,x∗=supx∗≤1Kx−∑i=1∞x,hifi,x∗=supx∗≤1Kx,x∗−∑i=1∞x,hifi,x∗=supx∗≤1x,K∗x∗−∑i=1∞x,hifi,x∗=supx∗≤1x,K∗x∗−x,∑i=1∞fi,x∗hi=supx∗≤1x,K∗x∗−x,THRFx∗=supx∗≤1x,K∗−THRFx∗≤supx∗≤1x·K∗−THRF·x∗≤K∗−THRF·x<x.
Thus,K−TFRH<1. Therefore, F,H is a pair of approximate K,K∗-dual Bessel sequences.Remark 25.
The above two theorems show that if a sequenceH=hii=1∞∈X∗ is close enough to a K-dual of the Xd∗-Bessel sequence F, it would become an approximate K-dual of F.



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