# Weaving Frames in HilbertC∗-Modules

Authors: Xin Zhao; Pengtong Li
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we investigate weaving frames in HilbertC∗-modules. We show that the equivalence of woven and weakly woven frames is still true for modular frames under certain conditions. By using the analysis operators of frames and frame operators of canonical duals, we obtain several perturbation results for given weaving frames and different weaving frame pairs. When the C∗-algebra is nonunital, we derive a correspondence of adjointable operators which is bounded below woven families. Finally, we discuss the redundancy of weaving frames in Hilbert C∗-modules.


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## Body

## 1. Introduction

Due to the useful applications in the characterization of function spaces, signal processing, and many other fields of applications, the theory of frames has developed rather rapidly in recent years. Various generalizations of frames have been developed. For example,g-frames [1], fusion frames [2], pseudoframes [3], and operator-valued frames [4]. Recently, Bemrose et al. [5] introduced a new concept of “weaving frames” in separable Hilbert spaces. This notion has potential applications in distributed signal processing and wireless sensor networks. Two frames xjj∈J and yjj∈J are said to be woven if there exist constants 0<C≤D such that, for every subset σ⊂J (J is a finite or countable index set), the family xjj∈σ∪yjj∈σc is a frame with frame bounds C,D. Over the years, various extensions of weaving frames have been investigated, see [6–10].By allowing the inner product to take values in aC∗-algebra, Hilbert C∗-modules are natural generalizations of Hilbert spaces. Note that the theory of Hilbert C∗-modules is quite different from that of Hilbert spaces. For example, not all bounded linear operators on Hilbert C∗-modules are adjointable. Thus, there are many essential differences between Hilbert space frames and modular frames. The problems about modular frames are more complicated than those in Hilbert spaces. In this paper, we investigate the weaving properties of modular fames. We refer to [11–13] for more information on frames in Hilbert C∗-modules.The paper is organized as follows. Section2 contains the definitions and some basic results about frames in Hilbert C∗-modules. In Section 3, we introduce the weaving frames in Hilbert C∗-modules and show that the equivalence of woven and weakly woven frames is still effective for Hilbert C∗-modules under certain conditions. Sufficient conditions for perturbations of weaving frames are given in Section 4. In Section 5, we give some weaving results for the nonunital case, and we investigate the redundancy property of weaving frames in Section 6.


## 2. Preliminaries

We first recall some basics about HilbertC∗-modules.Definition 1.
LetA be a C∗-algebra. A pre-Hilbert A-module ℋ is a linear space and an algebraic (left) A-module with compatible scalar multiplication: λax=λax=aλx for every λ∈ℂ, a∈A, and x∈ℋ, together with a map ⋅,⋅:ℋ×ℋ⟶A that possesses the following properties:(i)
x,x≥0, for every x∈ℋ(ii)
x,x=0 if and only if x=0(iii)
x,y=y,x∗, for every x, y∈ℋ(iv)
ax+by,z=ax,z+by,z, for every a, b∈A and x, y, z∈ℋThe map⋅:x↦x,x1/2 defines a norm on ℋ, and ℋ is called a Hilbert A-module if it is complete with respect to this norm. Denote by ℋ,ℋ⊂A the closure of the linear span of all x,y, x, y∈ℋ. A Hilbert A-module ℋ is called full if ℋ,ℋ=A.Letℋ and ℳ be Hilbert A-modules. A map F:ℋ⟶ℳ is called adjointable if there exists a map F∗:ℳ⟶ℋ such that(1)x,F∗y=Fx,y,x∈ℋ,y∈ℳ.Denote the set of all adjointable operators fromℋ to ℳ by ℒℋ,ℳ. The adjointable operator is automatically linear and bounded. In the case ℋ=K, we write ℒℋ for simplicity. For more information on Hilbert C∗-modules, see [14–18].Now, we review the definition of modular frames:Definition 2.
LetA be a unital C∗-algebra and J be a finite or countable index set. A sequence xjj∈J in a Hilbert A-module ℋ is called a (standard) frame if there exist constants C, D>0 such that, for every x∈ℋ,(2)Cx,x≤∑j∈Jx,xjxj,x≤Dx,x,where the sum in the middle of the inequality converges in norm. The constants C and D in (2) are called the lower and upper frame bounds, respectively. In this case, we say xjj∈J is a C,D-frame for convenience. Moreover, we call xjj∈J a Bessel sequence if only the right inequality in (2) holds; a C-tight frame if C=D and a Parseval frame if C=D=1.Recall that a HilbertA-module ℋ is finitely generated if there exists a finite set x1,…,xn⊂ℋ such that every element x∈ℋ can be expressed as an A-linear combination x=∑i=1naixi, ai∈A. A Hilbert A-module ℋ is countably generated if there exists a countable set of generators. It follows from Kasparov’s stabilization theorem (Theorem 2 in [14]) that every finitely or countable generated Hilbert C∗-module has a frame. Meanwhile, frames need not exist in general Hilbert C∗-modules (cf. [19]), so it is naturally restricting our consideration to finitely or countably generated Hilbert C∗-modules.Next, we give an equivalent definition of modular frames which is easy to be applied.Proposition 1 (see [13]).
Letℋ be a finitely or countably generated Hilbert A-module over a unital C∗-algebra A and X=xjj∈J⊆ℋ a sequence. Then, X=xjj∈J is a frame of ℋ with bounds (C, D) if and only if(3)Cx2≤∑j∈Jx,xjxj,x≤Dx2,for all x∈ℋ.For a unitalC∗-algebra A, let ℓ2A be a Hilbert A-module defined by(4)ℓ2A=ajj∈J∈A:∑j∈Jajaj∗converges in norm,with the inner product ajj∈J,bjj∈J=∑j∈Jajbj∗. Note that ℓ2A possesses a canonical basis ejj∈J, where ej takes value 1A (1Adenotes the unit element of A) at j and 0A everywhere else. For a given Bessel sequence X=xjj∈J in ℋ, its analysis operator,(5)UX:ℋ⟶ℓ2A,UXx=∑j∈Jx,xjej,is well defined and adjointable. The adjoint operator TX of UX fulfills TXej=xj for every j and is called the synthesis operator. By composing UX and TX, we obtain the frame operator SX:(6)SX:ℋ⟶ℋ,SXx=TXUXx=∑j∈Jx,xjxj,where SX is invertible and each x∈ℋ can be represented as(7)x=∑j∈Jx,SX−1xjxj,where SX−1xjj∈J is called the canonical dual frame of xjj∈J.At the end of this section, we present some fundamental results which will be used in the following sections.Proposition 2 (see [17]).
For an operatorF:ℋ⟶ℋ, the following conditions are equivalent:(i)
F is a positive element of ℒℋ(ii)
For everyx∈ℋ, Fx,x is positive in AProposition 3 (see [20]).
LetF∈ℒℋ. Then, for any x∈ℋ, we have Fx,Fx≤F2x,x.


## 3. Weaving Frames in HilbertC∗-Modules

In this section, we investigate the weaving frames in HilbertC∗-modules. It can be observed from (7) that each frame for a Hilbert C∗-module ℋ generates ℋ. This means that a Hilbert C∗-module that admits frames is necessarily countably generated.Throughout the rest of the paper, for ease of notation, let(8)m=1,…,m,for a given natural number m. First, we give the definition of weaving frames in Hilbert C∗-modules.Definition 3.
A finite family of framesxijj∈J,i∈m in a Hilbert A-module ℋ is said to be woven if there are universal constants C and D such that, for every partition σ1,…,σm of J, the family xiji=1,j∈σim=∪i∈mxijj∈σi is a frame for ℋ with lower and upper frame bounds C and D, respectively. In this case, we usually call xijj∈J,i∈m woven with universal bounds C,D. Each family xiji=1,j∈σim is called a weaving.It is known from Proposition 3.1 in [5] that every weaving for a woven family automatically has a universal upper frame bound. By the positivity of the summand in (2), we can also derive a universal Bessel bound for every weaving in a Hilbert C∗-module.Proposition 4.
Suppose thatxijj∈J,i∈m are Bessel sequences for a Hilbert A-module ℋ with Bessel bounds Di, respectively. Then, every weaving is a Bessel sequence with a Bessel bound ∑i=1mDi.Proof.
For every partitionσ1,…,σm of J and any x∈ℋ, we have(9)∑i=1m∑j∈σix,xijxij,x≤∑i=1m∑j∈Jx,xijxij,x≤∑i=1mDix,x,as required.
Consequently, we only need to check the lower frame bound when studying the property of woven frames. The following result implies that the woven property is preserved under some adjointable operators.Proposition 5.
Letℋ be a Hilbert A-module over a unital C∗-algebra A and xijj∈J,i∈m be a woven family of frames for ℋ with universal bounds C,D.(1)
IfF∈ℒℋis invertible, then the familyFxiji=1,j∈Jmis woven with universal boundsCF−1−2,DF2(2)
IfG∈ℒℋis a co-isometry, thenGxiji=1,j∈Jmis woven with universal boundsC,DProof.
(1)
For every partitionσ1,…,σm of J and any x∈ℋ,(10)CF∗x,F∗x≤∑i=1m∑j∈σix,FxijFxij,x≤DF∗x,F∗x.It follows from Proposition3 that(11)F−1−2x,x≤F∗x,F∗x≤F2x,x.Hence,Fxiji=1,j∈σim is a frame with bounds CF−1−2,DF2.(2) SinceG is an co-isometry, we know G∗x=x. Then, by E. C. Lance’s theorem [16], G∗x,G∗x=x,x. Now, the conclusion follows from a similar discussion of (1).
We show that multiplying the frames in the woven family by individual elements still preserves the woven property.Proposition 6.
Letℋ be a Hilbert A-module over a unital C∗-algebra A and xijj∈J,i∈m be a woven family of frames for ℋ with universal frame bounds C,D. If there exist aij∈A and 0<α≤β<∞ which satisfies α⋅1A≤aij∗aij≤β⋅1A, then aijxiji=1,j∈Jm is also woven with universal bounds αC,βD.Proof.
Recall thata≤b implies c∗ac≤c∗bc, for any a, b∈AsaAsa denotes the set of all self-adjoint elements in A and c∈A. On the one hand, we have for any partition σ1,…,σm of J and any x∈ℋ,(12)∑i=1m∑j∈σix,aijxijaijxij,x=∑i=1m∑j∈σix,xijai,j∗ai,jxij,x≤β∑i=1m∑j∈σix,xijxij,x≤βDx,x.
On the other hand, we also have(13)∑i=1m∑j∈σix,aijxijaijxij,x=∑i=1m∑j∈σix,xijai,j∗ai,jxij,x≥α∑i=1m∑j∈σix,xijxij,x≥αCx,x.
This gives the desired result.
Now, we give a definition of weakly woven frames.Definition 4.
A finite family of framesxijj∈J,i∈m in a Hilbert A-module ℋ is called weakly woven if, for every partition σ1,…,σm of J, the family xiji=1,j∈σim is a frame for ℋ.Bemrose et al. presented a characterization of the equivalence of woven and weakly woven frames in [5]. At the end of this section, we consider whether this equivalence is still effective for Hilbert C∗-modules. As a preparation, we need the following proposition.Proposition 7.
Letℋ be a Hilbert A-module over a unital C∗-algebra A and xijj∈J,i∈m be a family of frames for ℋ. Assume, for every partition Λii=1m of a finite subset of J and every ε>0, there exists a Λi⊂Γii∈m partition Γii=1m of J such that and the lower frame bound of xiji=1,j∈Γim is less than ε. Then, there exists a partition σii=1m of J such that xiji=1,j∈σim is not a frame.Proof.
Assume without loss of generality thatJ=ℕ. Choose ε=1/2 and Λi1=∅ for i∈m. Then, by assumption, there exists a partition Γi1i=1m of ℕ such that Λi1⊂Γi1 for all i∈m and the lower frame bound of xiji=1,j∈Γi1m is less than 1/2. So, there exists a element y1 such that(14)∑i=1m∑j∈Γi1y1,xijxij,y1<12y1,y1.
By the convergence of the series∑i=1m∑j∈ℕy1,xijxij,y1 for each x∈ℋ, we can find a k1∈ℕ such that ∑i=1m∑j≥k1y1,xijxij,y1<1/2y1,y1.
LetΛi2=Γi1∩k1 and ε=1/22. Then, there exists a partition Γi2i=1m of ℕ and an element y2∈ℋ such that Λi2⊂Γi2 for any i∈m and(15)∑i=1m∑j∈Γi2y2,xijxij,y2<122y2,y2.
Since∑i=1m∑j∈ℕy2,xijxij,y2 converges, there exists a k2>k1 such that(16)∑i=1m∑j≥k2y2,xijxij,y2<122y2,y2.
Continuing in this way, for a partitionΛin+1=Γin∩kn,i∈m of kn, we can find a partition Γin+1i=1m of ℕ and yn+1∈ℋ such that Λin+1⊂Γin+1 for every i∈m and(17)∑i=1m∑j∈Γin+1yn+1,xijxij,yn+1<12n+1yn+1,yn+1.
Similar to above, there exists akn+1>kn such that(18)∑i=1m∑j≥kn+1yn+1,xijxij,yn+1<12n+1yn+1,yn+1.
Now, setσi=∪n=1∞Λin+1 for i∈m and observe that σii=1m is a partition of ℕ. It follows from construction along with (17) and (18) that(19)∑i=1m∑j∈σiyn+1,xijxij,yn+1=∑i=1m∑j∈Λin+1yn+1,xijxij,yn+1+∑i=1m∑j∈σi∩kncyn+1,xijxij,yn+1≤∑i=1m∑j∈Γin+1yn+1,xijxij,yn+1+∑i=1m∑j≥kn+1yn+1,xijxij,yn+1<12nyn+1,yn+1,so that a lower frame bound of xiji=1,j∈σim is zero. Hence, it is not a frame and we have the result.Corollary 1.
Letℋ be a Hilbert A-module over a unital C∗-algebra A and xijj∈J,i∈m be a weakly woven family of frames for ℋ. Then, there exist disjoint finite sets Λ1,…Λm⊂J and a constant C>0 such that, for any partition σii=1m of J with Λi⊂σi for i∈m, the family xiji=1,j∈σim has a lower frame bound C.Every bounded sequence in a Hilbert space has a weakly convergent subsequence. However, this is usually not true for bounded sequences in general HilbertC∗-modules. Thus, the computations will be quite different than given in Proposition 4.5 in [5] and more conditions will be needed when investigating the analogous result.Theorem 1.
Letℋ be a full Hilbert A-module over a finite-dimensional C∗-algebra A and X=xjj∈J and Y=yjj∈J be frames for ℋ. Then, the following are equivalent:(i)
X and Y are woven(ii)
X and Y are weakly wovenProof.
(i)⇒ (ii) is clear. For the proof of (ii) ⇒ (i), we see from Proposition 4 that only a universal lower bound for X and Y needs to be shown. By Corollary 1, there exist disjoint finite sets Λ0, Γ0⊂J and a constant C>0 such that, for any subset σ0 of ℕ with Λ0⊂σ0 and Γ0⊂σ0c and every x∈ℋ,(20)∑j∈σ0x,xjxj,x+∑j∈σ0cx,yjyj,x≥Cx2.
Since permuting both frames simultaneously does not affect weaving, we permuteX and Y such that Λ0∪Γ0=m.
If we can prove that, for any partitionΛα,Γα of m, there exist Cα>0 such that, for every subset σα of ℕ with Λα⊂σα and Γα⊂σαc, the family xjj∈σα∪yjj∈σαc has a lower frame bound Cα; then, the frames X and Y are woven with a universal lower bound C0 = minCα:α, and we will obtain the desired result. Now, assume the above result is not true, and we will show that this yields a contradiction.
First, by hypothesis, we can find a partitionΛ1,Γ1 of m such that, for any ε>0, there exists a subset σ1 of ℕ with Λ1⊂σ1 and Γ1⊂σ1c such that the family xjj∈σ1∪yjj∈σ1c has a lower frame bound less than ε. Then, for all n∈ℕ, there exist subsets σn⊂ℕ with Λ1⊂σn and Γ1⊂σnc and hn∈ℋ with hn=1 such that(21)∑j∈σnhn,xjxj,hn+∑j∈σnchn,yjyj,hn<1n,and the sets σn satisfy the following properties:(1)
For everyk=1,2,…, either m+k∈σn, for all n≥k, or m+k∈σnc, for all n≥k(2)
There exist a subsetσ of ℕ with Λ1⊂σ and Γ1⊂σc such that m+k∈σ implies that m+k∈σn, for all n≥k, or if m+k∈σc, then m+k∈σnc, for all n≥k
Sinceℋ is a Hilbert C∗-module over a finite-dimensional C∗-algebra A, we see from Proposition 2.1 in [21] that, for the norm-bounded sequence hnn=1∞, there exists a subsequence hnii=1∞ of hnn=1∞ and h∈ℋ such that(22)y,hni−y,h⟶0asn⟶∞for anyy∈ℋ.
We reindexhni⟶hi and σni⟶σi and notice that (21), (1), and (2) are still satisfied in this construction. Now, fix k∈ℕ so that k>2/C, where C is the constant in (20). The fact that ℋ is full (Theorem 2.5 in [22]) and that finite-dimensional C∗-algebras are unital imply that hnn=1∞ converges in norm to h on finite-dimensional subspaces of ℋ. Hence, we can find an Nk∈ℕ such that, for all n≥Nk>k,(23)∑j∈m+kh−hn,xjxj,h−hn+∑j∈m+kh−hn,yjyj,h−hn<12k.
Denoteδ=Λ0∪σ∩mc and δn=Λ0∪σn∩mc. It follows from (20) that(24)∑j∈δnhn,xjxj,hn+∑j∈δnchn,yjyj,hn≥C.
By definition ofσ and δ, for n≥Nk>k, σ∩m,k=σn∩m,k, where m,k=m+1,m+2,…,m+k. Then, δ∩m+k=Λ0∪σ∩m,k=Λ0∪σn∩m,k=δn∩m+k, and thus, inequalities (21), (23), and (24) imply that(25)∑j∈δh,xjxj,h+∑j∈δch,yjyj,h≥∑j∈δ∩m+kh,xjxj,h+∑j∈δc∩m+kh,yjyj,h=∑j∈δn∩m+kh,xjxj,h+∑j∈δnc∩m+kh,yjyj,h≥12∑j∈δn∩m+khn,xjxj,hn+∑j∈δnc∩m+khn,yjyj,hn−∑j∈δn∩m+kh−hn,xjxj,h−hn+∑j∈δnc∩m+kh−hn,yjyj,h−hn≥12∑j∈δnhn,xjxj,hn+∑j∈δnchn,yjyj,hn−∑j∈δn∩m+kchn,xjxj,hn+∑j∈δnc∩m+kchn,yjyj,hn−∑j∈m+kh−hn,xjxj,h−hn+∑j∈m+kh−hn,yjyj,h−hn≥12∑j∈δnhn,xjxj,hn+∑j∈δnchn,yjyj,hn−12∑j∈δn∩m+kchn,xjxj,hn+∑j∈δnc∩m+kchn,yjyj,hn−12k≥12⋅C−12⋅1n−12k≥C2−1k>0.
This givesh≠0.
Finally, we will show thatxjj∈σ∪yjj∈σc is not a frame for ℋ. Similar to the above, for the set σ given in (2), σ∩m+k=σNk∩m+k. Thus, using (21) and (23),(26)∑j∈σ∩m+kh,xjxj,h+∑j∈σc∩m+kh,yjyj,h=∑j∈σNk∩m+kh,xjxj,h+∑j∈σNkc∩m+kh,yjyj,h≤2∑j∈σNk∩m+khNk,xjxj,hNk+∑j∈σNkc∩m+khNk,yjyj,hNk+2∑j∈σNk∩m+kh−hNk,xjxj,h−hNk+∑j∈σNkc∩m+kh−hNk,yjyj,h−hNk≤2∑j∈σNkhNk,xjxj,hNk+∑j∈σNkchNk,yjyj,hNk+2∑j∈m+kh−hNk,xjxj,h−hNk+∑j∈m+kh−hNk,yjyj,h−hNk≤2⋅1Nk+2⋅12k.
Lettingk⟶∞ in the above formula,(27)∑j∈σh,xjxj,h+∑j∈σch,yjyj,h=0,implying that xjj∈σ∪yjj∈σc is not a frame. Thus, a contradiction is met, concluding the proof.


## 4. Perturbations of Weaving Frames

Letℋ be a finitely or countably generated Hilbert A-module over a unital C∗-algebra A and X=xjj∈J be a Bessel sequence for ℋ with the analysis operator UX. To characterize weaving frames, we need to define the analysis operator relevant to the partitions. For any given subset σ of J, denoted by Pσ, the orthogonal projection is onto span¯ejj∈σ, where ejj∈J is the standard orthonormal basis of ℓ2A. Then, the analysis operator relevant to σ is defined by(28)UXσ:ℋ⟶ℓ2A,UXσx=PσUXx=∑j∈σx,xjej.Similarly, the synthesis operatorTXσ relevant to σ is given by(29)TXσ:ℓ2A⟶ℋ,TXσcjj∈J=TXPσcjj∈J=∑j∈σcjxj.By composingUXσ and TXσ, we obtain the frame operatorSXσ relevant to σ:(30)SXσ:ℋ⟶ℋ,SXσx=TXσUXσx=∑j∈σx,xjxj.LetX=xjj∈J and Y=yjj∈J be frames for ℋ. We compute, for any σ⊂J and any x∈ℋ,(31)UXσx+UYσcx2=∑j∈σx,xjxj,x+∑j∈σcx,yjyj,x.Then, by Proposition1, we know X and Y are woven with universal bounds C,D if and only if, for any σ⊂J,(32)Cx≤UXσx+UYσcx≤Dx.We begin investigating the perturbations of weaving frames.Theorem 2.
Letℋbe a HilbertA-module over a unitalC∗-algebraAandX=xjj∈J and Y=yjj∈Jbe frames forℋwith frame bounds (C1, D1) and (C2, D2), respectively. Assume that there exist constantsλ1, λ2, μ≥0such that(33)λ1C1+λ2D2+μ<C1,and for any x∈H,(34)UXx−UYx≤λ1UXx+λ2UYx+μx,whereUX and UYdenote the analysis operators forX and Y, respectively. Then,X and Yare woven frames with universal boundsC1−λ1C1−λ2D2−μ2and min D1+λ1D1+λ2D2+μ2,D1+D2.Proof.
For any givenσ⊂J, we denote ηjj∈J=xjj∈σc∪yjj∈σ and define an operator Uη:H⟶ℓ2A by(35)Uηx=UXσcx+UYσx=∑j∈σcx,xjej+∑j∈σx,yjej.
It is clear thatUη is the analysis operator for ηjj∈J. Hence, for any x∈ℋ,(36)UXx−Uηx=UXx−UXσcx−UYσx=∑j∈Jx,xjej−∑j∈σcx,xjej−∑j∈σx,yjej=∑j∈σx,xj−yjej=∑j∈σx,xj−yjxj−yj,x1/2≤∑j∈Jx,xj−yjxj−yj,x1/2=UXx−UYx.
On the one hand, we see from (34) that(37)UXx−Uηx≤UXx−Uηx≤λ1UXx+λ2UYx+μx.
Hence,(38)Uηx≥1−λ1UXx−λ2UYx−μx≥1−λ1C1x−λ2D2x−μx=C1−λ1C1−λ2D2−μx.
On the other hand, we also have(39)Uηx−UXx≤UXx−Uηx≤λ1UXx+λ2UYx+μx.
Therefore,(40)Uηx≤λ1+1UXx+λ2UYx+μx≤λ1+1D1x+λ2D2x+μx=D1+λ1D1+λ2D2+μx,as required.By choosingλ1=λ2=0 in Theorem 2, we see that X and Y are woven with universal bounds C1−μ2 and minD1+μ2,D1+D2, when μ<C1 and UXx−UYx≤μx. If the value for the universal bounds is irrelevant, we can get the following result.Corollary 2.
Letℋ be a Hilbert A-module over a unital C∗-algebra A and X=xjj∈J and Y=yjj∈J be frames for ℋ with frame bounds (C1, D1) and (C2, D2), respectively. If, for any x∈H,(41)UXx−UYx<C1x,orUXx−UYx<C2x,then X and Y are woven.The property of woven frames is preserved under an adjointable invertible operator. However, applying two different operators to woven frames does not always preserve the weaving property. Using Corollary2, we consider when the given frames are woven under different operators.Corollary 3.
Letℋ be a Hilbert A-module over a unital C∗-algebra A and X=xjj∈J and Y=yjj∈J be frames for ℋ with frame bounds (C1, D1) and (C2, D2), respectively. Assume F1, F2∈ℒℋ are invertible operators on ℋ. Then, F1xj and F2yj are woven frames when D1F1+D2F2F1−1<C1 or D1F1+D2F2F1−1<C2.Proof.
Note thatF1xjj∈J (resp. F2yjj∈J) is a frame with lower frame bound C1F1−1−2 (resp. C2F2−1−2) and analysis operator UXF1∗ (resp. UYF2∗). Thus, by assumption,(42)UXF1∗x−UYF2∗x≤UXF1∗x+UYF2∗x≤D1F1x+D2F2x<C1F1−1−2x.
Using Corollary2, we obtain the desired result.Canonical dual frames play a fundamental role in the study of woven frames. Using Corollary3, we now derive a perturbation result for canonical dual frames.Theorem 3.
Letℋ be a Hilbert A-module over a unital C∗-algebra A and X=xjj∈J and Y=yjj∈J be frames for ℋ with frame bounds (C1, D1) and (C2, D2), respectively. If, for any x∈H,(43)UXx−UYx<C1C2C2D1+D1D2+D2D1x,or(44)UXx−UYx<C1C2C1D2+D1D2+D2D1x,then SX−1xjj∈J and SY−1yjj∈J are woven.Proof.
Letμ denote C1C2/C2D1+D1D2+D2D1, and we have(45)UXx−UYx<μx,(46)μC1+μC1C2D1D2+D2=1D1.
Now, using (45), we get, for any x∈ℋ,(47)SXx−SYx=TXUXx−TYUYx≤TXUXx−TXUYx+TXUYx−TYUYx<D1UXx−UYx+μUYx<μD1+D2x.
Note thatSX−1xjj∈J and SY−1yjj∈J are frames with frame bounds 1/D1,1/C1 and 1/D2,1/C2, respectively. Hence, for any x∈ℋ,(48)SX−1x−SY−1x=SY−1SY−SXSX−1x<μC1C2D1+D2.
Now, denoteSX−1xjj∈J and SY−1yjj∈J by SX−1X and SY−1Y, respectively. Putting (45), (46), and (48) together, we get, for any x∈H,(49)USX−1Xx−USY−1Yx=∑j∈Jx,SX−1xjej−∑j∈Jx,SY−1yjej≤∑j∈Jx,SX−1xjej−∑j∈Jx,SX−1yjej+∑j∈Jx,SX−1yj−SY−1yjej=UXSX−1x−UYSX−1x+UYSX−1x−SY−1x<μSX−1x+D2SX−1x−SY−1x<μC1x+μC1C2D1D2+D2x=1D1x.
Now, the conclusion follows from Corollary3.
Canonical dual frames can be used in investigating the perturbation results of weaving frames.Proposition 8.
Letℋ be a Hilbert A-module over a unital C∗-algebra A and X=xjj∈J and Y=yjj∈J be frames for ℋ with frame bounds (C1, D1) and (C2, D2), respectively. If, for any x∈H,(50)∑j∈Jxj−yj⋅SX−1xj=μ<1,then xjj∈J and yjj∈J are woven frames with universal bounds C11−μ2,D1+D2.Proof.
For anyσ⊂J, we denote ηjj∈J=xjj∈σc∪yjj∈σ and define an operator L:ℋ⟶ℋ by(51)Lx=∑j∈Jx,SX−1xjηj.
Note thatSX−1xj is a 1/D,1/C-frame. For any x∈ℋ, we compute(52)x−Lx=∑j∈Jx,SX−1xjxj−ηj=∑j∈σx,SX−1xjxj−yj≤∑j∈σxj−yj⋅SX−1xj⋅x=μx.
Hence, (50) implies that L is invertible and(53)L≤1+μ,L−1≤11−μ.
Everyx∈ℋ can be written as(54)x=LL−1x=∑j∈JL−1x,SX−1xjηj,implying that(55)x4=x,x2=∑j∈JL−1x,SX−1xjηj,x2≤∑j∈JL−1x,SX−1xjSX−1xj,L−1x⋅∑j∈Jx,ηjηj,x≤1C1L−1x2⋅∑j∈Jx,ηjηj,x≤1C11−μ2x2⋅∑j∈Jx,ηjηj,x.
Hence, we have obtained the claimed lower frame bound:(56)∑j∈Jx,ηjηj,x≥C11−μ2x2,as required.We have known frames that small perturbations of each other are woven. In this section, we consider the sufficient conditions under which every perturbation of a given woven frame pair (xjj∈J, yjj∈J) is still woven.Theorem 4.
Letℋ be a Hilbert A-module over a unital C∗-algebra A. Suppose X=xjj∈J and Y=yjj∈J are woven frames with universal bounds M,N and W=wjj∈J and Z=zjj∈J are frames for ℋ. Assume that there exist μ, μ′≥0 such that(57)λ1M+λ1′M+μ+μ′<M,and for any x∈ℋ, we have(58)UXx−UWx≤λ1UXx+λ2UWx+μx,(59)UYx−UZx≤λ1′UYx+λ2′UZx+μ′x.Then,W and Z are woven with lower universal bound M−λ1M−λ1′M−μ−μ′/1+λ2+λ2′2.Proof.
For any givenσ⊂J, we denote ηjj∈J=xjj∈σc∪yjj∈σ and ξjj∈J=wjj∈σc∪zjj∈σ. Then, by assumption, we know ηj is a M,N-frame.
It follows fromUXσ=PσUη that UXσx≤Uηx. Hence, using (58) and (59),(60)Uξx−Uηx≤Uξx−Uηx≤UZσx+UWσcx−UYσx−UXσcx≤UZσx−UYσx+UMσcx−UXσcx≤λ1UZσx+λ2UYσx+μx+λ1′UWσcx+λ2′UXσcx+μ′x≤λ1Uξx+λ2Uηx+μx+λ1′Uξx+λ2′Uηx+μ′x=λ1+λ1′Uξx+λ2+λ2′Uηx+μ+μ′x.
Therefore,(61)Uηx≥M−λ1M−λ1′M−μ−μ′1+λ2+λ2′x,and we have the result.Theorem 5.
Letℋ be a Hilbert A-module over a unital C∗-algebra A. Suppose X=xjj∈J and Y=yjj∈J are woven frames with universal bounds M,N and W=wjj∈J and Z=zjj∈J are frames for ℋ. Assume that there exist μ, μ′≥0 such that μ+μ′<1, and for any x∈ℋ,(62)∑j∈Jxj−wj⋅SX−1xj=μ,∑j∈Jyj−zj⋅SY−1yj=μ′.Then,W and Z are woven with lower universal bounds M1−μ−μ′2.Proof.
SinceX and Y are woven with universal bounds M,N, we obtain that ηjj∈J=xjj∈σc∪yjj∈σ is a M,N-frame for any σ⊂J. Now, we denote ξjj∈J=wjj∈σc∪zjj∈σ and define an operator L:ℋ⟶ℋ by(63)Lx=∑j∈Jx,Sη−1ηjξj.
Then, for assumption, we have, for anyx∈ℋ,(64)x−Lx=∑j∈Jx,Sη−1ηjηj−ξj≤∑j∈σx,SY−1yjyj−zj+∑j∈σcx,SX−1xjxj−wj≤∑j∈σyj−zj⋅SY−1yj⋅x+∑j∈σcxj−wj⋅SX−1xj⋅x≤μ+μ′x.
It follows thatL is invertible and(65)L−1≤11−μ−μ′.
Now, for anyx∈ℋ,(66)x=LL−1x=∑j∈JL−1x,Sη−1ηjξj.
Therefore,(67)x4=x,x2=∑j∈JL−1x,Sη−1ηjξj,x2≤∑j∈JL−1x,Sη−1ηjSη−1ηj,L−1x⋅∑j∈Jx,ξjξj,x≤1ML−1x2⋅∑j∈Jx,ξjξj,x.
Putting (65) and (66) together, we have(68)∑j∈Jx,ηjηj,x≥M1−μ−μ′2x2,as required.


## 5. Weaving Frames in HilbertC∗-Modules over Nonunital C∗-Algebras

The frames for a Hilbert spaceℋ are precisely the families Tejj∈J, where T:ℓ2J⟶ℋ is a bounded surjection and ejj∈J is the canonical basis of ℓ2J. This correspondence is still true for modular frames when the C∗-algebra A is unital. Assume xjj∈J is a frame for a Hilbert C∗-module over a unital C∗-algebras with analysis operator U. We can derive from Corollary 2 that any adjointable operators belonging to a small neighborhood of U determine a frame which is woven with xjj∈J. However, for the nonunital case, the situation is more complicated. Let A be an arbitrary C∗-algebra and ℋ be a Hilbert A-module. An adjointable surjection from ℓ2A to ℋ may not be the analysis operator for any frame. Furthermore, there exist operator whose any neighborhood can contain an operator that is not the analysis operator for any frame. Thus, we need a new notation to obtain analogues of the unital case.Arambasic et al. introduced the concept of the outer frame in [23] to investigate the bijective correspondence of all surjections from ℓ2A to ℋ and all both frames or outer frames for ℋ. Let Mℋ be the multiplier module of ℋ. Note that Mℋ is a Hilbert MA-module which contains ℋ as an ideal submodule. Recall that a sequence xjj∈J in Mℋ is called an outer frame for ℋ if xi∈Mℋ\ℋ for at least one i∈J, and (2) holds for every x∈ℋ and suitable positive constants (C, D). Let X=xjj∈J and Y=yjj∈J be frames or outer frames for ℋ, and we say X and Y are woven if there are universal constants C and D such that, for any subset σ of J, xjj∈σc∪yjj∈σ is a frame or an outer frame for ℋ with bounds (C, D) (Theorem 3.19 in [23]) shows that, for any surjection T∈ℒℓ2A,ℋ, there exists a unique frame or an outer frame xjj∈J whose synthesis operator coincides with T. Now, we give the key result of this section.Theorem 6.
Letℋ be a Hilbert A-module over a C∗-algebra A and U, V∈ℒℓ2A,ℋ. Suppose U is bounded below and(69)U−V<maxU∗U−1/2−1,V∗V−1/2−1.Then,U and V are analysis operators for a woven family of frames or outer frames.Proof.
Without loss of generality, we assumeU−V<U∗U−1/2−1. Since U is bounded below,(70)U∗U−1/2−1=minm≥0:U∗U−1/2x≤mx,x∈ℋ−1=minm≥0:x≤mU∗U1/2x,x∈ℋ−1=minm≥0:1mx≤Ux,x∈ℋ−1=maxm≥0:mx≤Ux,x∈ℋ.
Writeμ:=U−V. Then, by assumption,(71)Vx≥Ux−Ux−Vx≥U∗U−1/2−1−μx,∀x∈ℋ,implying that V is bounded below. Hence, U and V are analysis operators of frames or outer frames denoted by xjj∈J and yjj∈J, respectively.
For anyσ⊂J, we denote ηjj∈J=xjj∈σc∪yjj∈σ and define an operator Uη:ℋ⟶ℓ2A by(72)Uηx=∑j∈σcx,xjej+∑j∈σx,yjej.
Then, for anyx∈ℋ,(73)Uηx−Ux=∑j∈σx,yj−xjej≤Ux−Vx.
It follows that(74)Uηx≥Ux−Uηx−Ux≥U∗U−1/2−1−mx.
Thus,ηjj∈J is a frame or an outer frame for ℋ. We can prove the result similarly using U−V<V∗V−1/2−1, and we have the result.WhenA is unital, Mℋ=ℋ. There are no outer frames for ℋ, and clearly, Theorem 6 holds for ordinary frames in the Hilbert A-module.


## 6. Redundancy of Weaving Frames

In this section, we study the redundancy of frames in HilbertC∗-modules. Recall that a frame xjj∈J for a Hilbert A-module ℋ is said to be a (standard) Riesz basis for ℋ if it satisfies the following:(i)
xj≠0, for all j(ii)
If anA-linear combination ∑j∈Scjxj with coefficients cj:j∈S⊂A and S⊂J is equal to zero, then every summand cjxj is zeroIt is shown in [5] that a frame which is woven with a Riesz basis in a Hilbert space must be a Riesz basis and [24] shows that woven frames have the same excess. However, the redundancy property of modular frames is quite different from that of frames in Hilbert spaces. As was pointed out in Section 2, every countably generated Hilbert C∗-module admits a (standard) frame. However, there exist countably generated Hilbert C∗-modules that contain no orthonormal basis, even a Riesz basis, cf. [25].We first state the following characterization for Riesz bases in HilbertA-modules.Proposition 9 (see [12]).
Suppose thatxjj∈J is a frame of ℋ; then, xjj∈J is a Riesz basis if and only if(i)
xj≠0 for all j∈J(ii)
∑j∈Jcjxj=0 for some sequence cjj∈J∈ℓ2A, then cjxj=0, for all jNow, we derive the following result.Theorem 7.
Letℋ be a Hilbert A-module over a unital C∗-algebra A and X=xjj∈J and Y=yjj∈J be woven frames for ℋ. If X is a Riesz basis with ranθX=ℓ2A, then Y is a Riesz basis with ranθY=ℓ2A.Proof.
Letejj∈J be the canonical basis of ℓ2A. Note that every modular frame can be written as Tejj∈J for a surjective adjointable operator T. Since ranθX=ℓ2A, the synthesis operator TX is invertible. Then, by Proposition 5, we can assume that X=ejj∈J and Y=Tejj∈J, where T is a surjective operator in ℒℓ2A.
By way of contradiction, assume thatranθY≠ℓ2A. This means that kerTY≠0. Then, there exists cjj∈J with ci≠0 for some i, such that ∑j∈JcjTej=0. This implies that Tei=∑j≠i−cj/ciTej, and thus, Tei∈span¯Tejj≠i. Without loss of generality it may be assumed that Te1∈span¯Tejj=2∞. For given any ε>0, there exists n∈ℕ such that dTe1,spanTejj=2n<ε. Now, choose x∈spanejj=1n∩spanTejj=2n⊥ with x=1. Then, for any y∈spanTejj=2n, we compute(75)∑j=1nx,ej2+∑j=n+1∞x,Tej2=x,Te12=x,Te1−y2≤Te1−y2<ε.
So,ejj=1n∪Tejj=n+1∞ is not a frame. Then, we have ranθY=ℓ2A. Hence, TYcjj∈J=0 can imply cj=0 for all j. Thus, cjxj=0 for all j, and it follows from Proposition 9 that Y is a Riesz basis.
Theorem 4.15 in [23] proved that a frame or an outer frame for a full Hilbert A-module has a unique dual if and only if the analysis operator is surjective. Combined with Theorem 7, we have the following.Corollary 4.
Letℋ be a full Hilbert A-module and X=xjj∈J and Y=yjj∈J be a woven family of frames or outer frames. If X has a unique dual, then Y has a unique dual.Now, we derive a sufficient condition for the situation that a frame which is woven to a Riesz basis is a Riesz basis itself.Proposition 10.
Letℋ be a Hilbert A-module over a unital C∗-algebra A and X=xjj∈J and Y=yjj∈J be frames for ℋ with frame bounds (C1, D1) and (C2, D2), respectively. Assume that X is a Riesz basis, and there exist constants λ1, λ2≥0 with(76)λ1D1+λ2D2<minC1,C2,such that, for any cjj∈J∈ℓ2A,(77)TXcjj∈J−TYcjj∈J≤λ1TXcjj∈J+λ2TYcjj∈J,where TX and TY are the synthesis operators for X and Y, respectively. Then, X and Y are woven frames and Y is a Riesz basis.Proof.
Define an operatorL and a sequence ηjj∈J in the same way as the proof of Theorem 9. Using (77), on the sequence cji∈J:=x,SX−1xjj∈J, we obtain that(78)x−Lx=∑j∈Jx,SX−1xjxj−ηj=∑j∈σcjxj−yj≤λ1TXσcjj∈J+λ2TYσcjj∈J≤λ1TX⋅cjj∈J+λ2TY⋅cjj∈J≤λ1D1+λ2D2cjj∈J≤λ1D1+λ2D2C1x.
Hence, (76) implies that L is invertible and L−1≤C1/C1−λ1D1−λ2D2. Every x∈ℋ can be written as(79)x=LL−1x=∑j∈JL−1x,SX−1xjηj,implying that(80)x4=x,x2=∑j∈JL−1x,SX−1xjηj,x2≤∑j∈JL−1x,SX−1xjSX−1xj,L−1x⋅∑j∈Jx,ηjηj,x≤1C1L−1x2⋅∑j∈Jx,ηjηj,x.
Therefore,(81)∑j∈Jx,ηjηj,x≥C1−λ1D1−λ2D22x2,and thus, we get X and Y are woven.
Now, let us proveY is a Riesz basis. Assume yi=0 for some i. It follows from (76) that λ1, λ2<1 and xi≤λ1xi. So, xi=0, and it is a contradiction. Hence, yj≠0, for each j∈J. Now, let cjj∈J∈kerTX. Using (77), we obtain that TYcjj∈J≤λ2TYcjj∈J, which implies that cjj∈J∈kerTY. So, ranθX=ranθY, and it follows from Theorem 3.1 in [12] that Y is a Riesz basis.



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