Weaving Frames in Hilbert C ∗-Modules

In this paper, we investigate weaving frames in Hilbert C∗-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.


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. is notion has potential applications in distributed signal processing and wireless sensor networks. Two frames x j j∈J and y j j∈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 x j j∈σ ∪ y j j∈σ c is a frame with frame bounds (C, D). Over the years, various extensions of weaving frames have been investigated, see [6][7][8][9][10].
By allowing the inner product to take values in a C * -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. us, there are many essential differences between Hilbert space frames and modular frames. e 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][12][13] for more information on frames in Hilbert C * -modules. e paper is organized as follows. Section 2 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.
(ii) 〈x, x〉 � 0 if and only if x � 0 (iii) 〈x, y〉 � 〈y, x〉 * , for every x, y ∈ H (iv) 〈ax + by, z〉 � a〈x, z〉 + b〈y, z〉, for every a, b ∈ A and x, y, z ∈ H e map ‖ · ‖: x ↦ ‖〈x, x〉‖ 1/2 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.
Let H and M be Hilbert (1) Denote the set of all adjointable operators from H to M by L(H, M). e adjointable operator is automatically linear and bounded. In the case H � K, we write L(H) for simplicity. For more information on Hilbert C * -modules, see [14][15][16][17][18]. Now, we review the definition of modular frames: Definition 2. Let A be a unital C * -algebra and J be a finite or countable index set. A sequence x j j∈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, where the sum in the middle of the inequality converges in norm. e constants C and D in (2) are called the lower and upper frame bounds, respectively. In this case, we say x j j∈J is a (C, D)-frame for convenience. Moreover, we call x j j∈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 Hilbert A-module H is finitely generated if there exists a finite set x 1 , . . . , x n ⊂ H such that every element x ∈ H can be expressed as an A-linear combination x � n i�1 a i x i , a i ∈ A. A Hilbert A-module H is countably generated if there exists a countable set of generators. It follows from Kasparov's stabilization theorem ( eorem 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]). Let H be a finitely or countably generated Hilbert A-module over a unital C * -algebra A and X � x j j∈J ⊆ H a sequence. en, X � x j j∈J is a frame of H with bounds (C, D) if and only if for all x ∈ H.
For a unital C * -algebra A, let ℓ 2 (A) be a Hilbert A-module defined by with the inner product 〈 a j j∈J , b j j∈J 〉 � j∈J a j b * j . Note that ℓ 2 (A) possesses a canonical basis e j j∈J , where e j takes value 1 A (1 A denotes the unit element of A ) at j and 0 A everywhere else. For a given Bessel sequence X � x j j∈J in H, its analysis operator, is well defined and adjointable. e adjoint operator T X of U X fulfills T X e j � x j for every j and is called the synthesis operator. By composing U X and T X , we obtain the frame operator S X : where S X is invertible and each x ∈ H can be represented as where S − 1 X x j j∈J is called the canonical dual frame of x j j∈J .
At the end of this section, we present some fundamental results which will be used in the following sections.

Weaving Frames in Hilbert C * -Modules
In this section, we investigate the weaving frames in Hilbert C * -modules. It can be observed from (7) that each frame for a Hilbert C * -module H generates H.
is means that a Hilbert C * -module that admits frames is necessarily countably generated.
roughout the rest of the paper, for ease of notation, let for a given natural number m. First, we give the definition of weaving frames in Hilbert C * -modules. woven with universal bounds (C, D). Each family 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.
as required.
Consequently, we only need to check the lower frame bound when studying the property of woven frames. e following result implies that the woven property is preserved under some adjointable operators.

Proof
(1) For every partition σ 1 , . . . , σ m of J and any x ∈ H, It follows from Proposition 3 that Hence, Fx ij m i�1,j ∈ σ i is a frame with bounds (2) Since G is an co-isometry, we know ‖G * x‖ � ‖x‖. en, 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. Let H be a Hilbert A-module over a unital C * -algebra A and x ij j∈J , i ∈ [m] be a woven family of Proof. Recall that a ≤ b implies c * ac ≤ c * bc, for any a, b ∈ A sa (A sa denotes the set of all self-adjoint elements in A) and c ∈ A. On the one hand, we have for any partition On the other hand, we also have is gives the desired result.
Now, we give a definition of weakly woven frames. □ 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.
Proof. Assume without loss of generality that J � N.
By the convergence of the series Since m i�1 j∈N 〈y 2 , x ij 〉〈x ij , y 2 〉 converges, there exists a k 2 > k 1 such that Continuing in this way, for a partition and observe that σ i m i�1 is a partition of N. It follows from construction along with (17) and (18) so that a lower frame bound of x ij m i�1,j ∈ σ i is zero. Hence, it is not a frame and we have the result. Every bounded sequence in a Hilbert space has a weakly convergent subsequence. However, this is usually not true for bounded sequences in general Hilbert C * -modules. us, 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. Let H be a full Hilbert
A-module over a finitedimensional C * -algebra A and X � x j j∈J and Y � y j j∈J be frames for H. en, the following are equivalent: (i) X and Y are woven (ii) X and Y are weakly woven Proof. (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 Λ 0 , Γ 0 ⊂ J and a constant C > 0 such that, for any subset σ 0 of N with Λ 0 ⊂ σ 0 and Γ 0 ⊂ σ c 0 and every x ∈ H, Since permuting both frames simultaneously does not affect weaving, we permute X and Y such that Λ 0 ∪ Γ 0 � [m].
If we can prove that, for any partition Λ α , Γ α of [m], there exist C α > 0 such that, for every subset σ α of N with Λ α ⊂ σ α and Γ α ⊂ σ c α , the family x j j∈σ α ∪ y j j∈σ c α has a lower frame bound C α ; then, the frames X and Y are woven with a universal lower bound C 0 � min C α : 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 partition Λ 1 , Γ 1 of [m] such that, for any ε > 0, there exists a subset σ 1 of N with Λ 1 ⊂ σ 1 and Γ 1 ⊂ σ c 1 such that the family x j j∈σ 1 ∪ y j j∈σ c 1 has a lower frame bound less than ε. en, for all n ∈ N, there exist subsets σ n ⊂ N with Λ 1 ⊂ σ n and Γ 1 ⊂ σ c n and h n ∈ H with ‖h n ‖ � 1 such that and the sets σ n satisfy the following properties: (1) For every k � 1, 2, . . ., either m + k ∈ σ n , for all n ≥ k, or m + k ∈ σ c n , for all n ≥ k (2) ere exist a subset σ of N with Λ 1 ⊂ σ and Γ 1 ⊂ σ c such that m + k ∈ σ implies that m + k ∈ σ n , for all n ≥ k, or if m + k ∈ σ c , then m + k ∈ σ c n , for all n ≥ k Since H 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 h n ∞ n�1 , there exists a subsequence h n i ∞ i�1 of h n ∞ n�1 and h ∈ H such that 〈y, h n i 〉 − 〈y, h〉 � � � � � � � � � � ⟶ 0(as n ⟶ ∞) for any y ∈ H.

(22)
We reindex h n i ⟶ h i and σ n i ⟶ σ i 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). e fact that H is full ( eorem 2.5 in [22]) and that finitedimensional C * -algebras are unital imply that h n ∞ n�1 converges in norm to h on finite-dimensional subspaces of H. Hence, we can find an N k ∈ N such that, for all n ≥ N k > k, is gives h ≠ 0. Finally, we will show that x j j∈σ ∪ y j j∈σ c is not a frame for H. Similar to the above, for the set σ given in (2), us, using (21) and (23), j∈σ∩ [m+k] 〈h, x j 〉〈x j , h〉 + j∈σ c ∩ [m+k] 〈h, y j 〉〈y j , h〉 〈h, y j 〉〈y j , h〉 implying that x j j∈σ ∪ y j j∈σ c is not a frame. us, a contradiction is met, concluding the proof.

Perturbations of Weaving Frames
Let H be a finitely or countably generated Hilbert A-module over a unital C * -algebra A and X � x j j∈J be a Bessel sequence for H with the analysis operator U X . To characterize weaving frames, we need to define the analysis operator relevant to the partitions. For any given subset σ of J, denoted by P σ , the orthogonal projection is onto span e j j∈σ , where e j j∈J is the standard orthonormal basis of ℓ 2 (A). en, the analysis operator relevant to σ is defined by Similarly, the synthesis operator T σ X relevant to σ is given by By composing U σ X and T σ X , we obtain the frame operator S σ X relevant to σ: Let X � x j j∈J and Y � y j j∈J be frames for H. We compute, for any σ ⊂ J and any x ∈ H, en, by Proposition 1, we know X and Y are woven with universal bounds (C, D) if and only if, for any σ ⊂ J, We begin investigating the perturbations of weaving frames.

Theorem 2.
Let H be a Hilbert A-module over a unital C * -algebra A and X � x j j∈J and Y � y j j∈J be frames for H with frame bounds (C 1 , D 1 ) and (C 2 , D 2 ), respectively. Assume that there exist constants λ 1 , λ 2 , μ ≥ 0 such that and for any x ∈ H, where U X and U Y denote the analysis operators for X and Y, respectively. en, X and Y are woven frames with universal bounds Proof. For any given σ ⊂ J, we denote η j j∈J � x j j∈σ c ∪ y j j∈σ and define an operator U η : H ⟶ ℓ 2 (A) by It is clear that U η is the analysis operator for η j j∈J . Hence, for any x ∈ H, (36) On the one hand, we see from (34) that Hence, On the other hand, we also have erefore, as required.

Corollary 2.
Let H be a Hilbert A-module over a unital C * -algebra A and X � x j j∈J and Y � y j j∈J be frames for H with frame bounds (C 1 , D 1 ) and (C 2 , D 2 ), respectively. If, for any x ∈ H, then X and Y are woven.
e 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 Corollary 2, we consider when the given frames are woven under different operators.

Corollary 3. Let H be a Hilbert
A-module over a unital C * -algebra A and X � x j j∈J and Y � y j j∈J be frames for H with frame bounds (C 1 , D 1 ) and (C 2 , D 2 ), respectively. Assume F 1 , F 2 ∈ L(H) are invertible operators on H. en, F 1 x j and F 2 y j are woven frames when ( �� � Proof. Note that F 1 x j j∈J (resp. F 2 y j j∈J ) is a frame with lower frame bound C 1 ‖F − 1 1 ‖ − 2 (resp. C 2 ‖F − 1 2 ‖ − 2 ) and analysis operator U X F * 1 (resp. U Y F * 2 ). us, by assumption, Using Corollary 2, we obtain the desired result.

□
Canonical dual frames play a fundamental role in the study of woven frames. Using Corollary 3, we now derive a perturbation result for canonical dual frames.

Theorem 3.
Let H be a Hilbert A-module over a unital C * -algebra A and X � x j j∈J and Y � y j j∈J be frames for H with frame bounds (C 1 , D 1 ) and (C 2 , D 2 ), respectively. If, for any x ∈ H, or then S − 1 X x j j∈J and S − 1 Y y j j∈J are woven.
Now, using (45), we get, for any x ∈ H, Note that S − 1 X x j j∈J and S − 1 Y y j j∈J are frames with frame bounds (1/D 1 , 1/C 1 ) and (1/D 2 , 1/C 2 ), respectively. Hence, for any x ∈ H, Now, denote S − 1 X x j j∈J and S − 1 Y y j j∈J by S − 1 X X and S − 1 Y Y, respectively. Putting (45), (46), and (48) together, we get, for any x ∈ H, Now, the conclusion follows from Corollary 3. Canonical dual frames can be used in investigating the perturbation results of weaving frames.
Proof. For any σ ⊂ J, we denote η j j∈J � x j j∈σ c ∪ y j j∈σ and define an operator L: H ⟶ H by Note that S − 1 X x j is a (1/D, 1/C)-frame. For any x ∈ H, we compute Hence, (50) implies that L is invertible and Every x ∈ H can be written as implying that Hence, we have obtained the claimed lower frame bound: j∈J 〈x, η j 〉〈η j , x〉 � � � � � � � � � � � � � � � � � � � � 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 ( x j j∈J , y j j∈J ) is still woven.