Relay fusion frames in Banach spaces

Guoqing Hong; Pengtong Li

doi:10.1515/math-2022-0554

Article Open Access

Relay fusion frames in Banach spaces

Guoqing Hong and Pengtong Li

Published/Copyright: February 14, 2023

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$Open Mathematics$

From the journal Open Mathematics Volume 21 Issue 1

Abstract

The relay fusion frames have been recently introduced in Hilbert spaces to model sensor relay networks and distributed sensor relay systems, which are deeply connected with compressed sensing. In this article, we introduce the notions of relay fusion frames and Banach relay fusion frames in Banach spaces and study certain attractive properties of relay fusion frames in this more general setting. In a particular sense, Schauder frames can be shown to be a special case of relay fusion frames. Moreover, the stability issue of relay fusion frames and Banach relay fusion frames will be addressed.

Keywords: relay fusion frame; Banach relay fusion frame; perturbation; Banach space

MSC 2010: 42C15; 46B15

1 Introduction

The concept of frames in a separable Hilbert space was originally introduced by Duffin and Schaeffer in the context of the non-harmonic Fourier series [1]. They satisfy the well-known property of perfect reconstruction, in that any vector of the Hilbert space can be synthesized back from its inner products with the frame vectors [2]. Fusion frames were launched by Casazza and Kutyniok in [3] and further developed in their joint article [4] with Li et al. Fusion frames are an emerging topic of frame theory with applications to communications and distributed processing. Relay fusion frames, or simply, R-fusion frames, are recent developments of fusion frames and g-frames that provide a natural mathematical framework for three-stage (or, more generally, hierarchical) data processing. The notion of an R-fusion frame was introduced in [5] with the main idea already presented in [6] and has become a tool in the implementation of distributed relay systems and distributed sparse recovery [7]. The main purpose of this article is to introduce some of the natural generalizations of R-fusion frame theory to Banach spaces and study some important properties and stability of R-fusion frames in this more general setting.

As well known, the extension of frame theory to Banach Spaces is an important issue, not only from a theoretical perspective but also from a number of applications involving signals in Banach Spaces [8–10]. Mathematically, the extension of frame theory to Banach spaces is highly nontrivial and is deeply related to some well-known problems in Banach space theory. The classical frames have been extended to Banach spaces in a variety of ways. One way is to extend the frames of Banach spaces to the compression of unconditional bases of larger Banach spaces [11,12]. This definition captures the unconditional convergence property of frame operators, which seems to be the core of Hilbert space frame theory. Another way is to generalize the frame inequality, such as [13,14]. The third method is to generalize the dual frame pairs in Hilbert spaces and the Schauder bases in Banach spaces [15,16]. Various frames for Banach spaces have played a key role in the development of wavelet theory and Gabor or Fourier analysis. In addition to the theoretical appeal, the frames in the L p spaces and other Banach function spaces are effective tools for modeling various natural signals and images [17,18]. In general, unlike Hilbert Spaces, Banach spaces do not necessarily have inner product and orthogonality, and most Banach Spaces do not have reflexivity, so it is relatively difficult to study frames in Banach Spaces. One way to overcome these difficulties is to use operator theory in Banach Spaces [19].

R-fusion frame is a new signal representation method that uses collections of relay subspaces instead of vectors to represent signals. Such a representation provides significant flexibility compared to classical frame representations. In this article we define the concept of R-fusion frames in Banach spaces and make a detailed study of R-fusion frames in this more general setting. The rest of the article is organized as follows: In Section 2, we give the basic definitions and certain preliminaries to be used in the article. R-fusion frames for Banach spaces will be introduced in Section 3, and some basic properties about R-fusion frames in this context will be developed in this section. The investigation of stability of R-fusion frames in Banach spaces is in Section 4.

2 Preliminaries

We first give the background material needed for the article. For simplicity, a set { α i } i ∈ I indexed by I will be abbreviated as { α i } throughout the article, where I is a countable index set. Let X , Y , Y i be Banach spaces and X ∗ be the dual space of X . The set of all bounded operators from X to Y i is denoted by ℬ ( X , Y i ) and ℬ ( X , X ) is abbreviated as ℬ ( X ) . For completeness, we review the definitions of the BK-spaces and various frames in Banach spaces, and analyze the relationships among various frames in Banach spaces. We start with recalling the definition of a BK-space.

Definition 1

Let X d be a Banach space of scalar-valued sequences. It is called a BK-space if the coordinate functionals are continuous on X d , and it is solid if whenever { b i } and { c i } are sequences with { c i } ∈ X d and ∣ b i ∣ ≤ ∣ c i ∣ , then { b i } ∈ X d and ‖ { b i } ‖ ≤ ‖ { c i } ‖ .

According to the result presented in [20], the dual space of a BK-space is also a BK-space with the duality relation:

d ( c ) = ∑ i ∈ I c i d i , ∀ c = { c i } ∈ X d , d = { d i } ∈ X d ∗ .

Example 1

Let c 0 be the space of sequences converging to zero and c be the space of convergent sequences. Then c 0 and c are solid BK-spaces. Moreover, we note that every ℓ p ( 1 ≤ p ≤ ∞ ) is a solid BK-space.

We will need to work with X d -frames.

Definition 2

Let X be a Banach space and X d be a BK-space. A countable collection of vectors { g i } in X ∗ is called an X d -frame for X , if

{ g i ( x ) } ∈ X d for all x ∈ X ,
the norms ‖ x ‖ and ‖ { g i ( x ) } ‖ are equivalent, i.e., there exist constants 0 < α ≤ β < ∞ such that

(2.1) α ‖ x ‖ ≤ ‖ { g i ( x ) } ‖ ≤ β ‖ x ‖ .

In this case, we say that { g i } is an ( α , β ) - X d -frame for X .

If X is a Hilbert space and X d = ℓ 2 ( I ) , (2.1) means that { g i } is a frame, and in this case, it is well known that there exists a sequence { f i } in X such that

f = ∑ ⟨ x , f i ⟩ g i = ∑ ⟨ x , g i ⟩ f i , x ∈ X .

Similar reconstruction formulas are not always available in the Banach space setting. This is the reason behind the following definition [14].

Definition 3

Let X be a Banach space and X d be a sequence space. Given a bounded linear operator S : X d → X and an X d -frame { g i } , we say that ( { g i } , S ) is a Banach frame for X with respect to X d if for all x ∈ X ,

(2.2) S ( { g i ( x ) } ) = x .

In the following example, we construct a new Banach frame for ℓ p with respect to ℓ p ( 1 < p < ∞ ) .

Example 2

Let X = X d = ℓ p for p ∈ ( 1 , ∞ ) and k ∈ Z , 1 ≤ δ < ∞ . Let { e i } i ∈ Z be the standard basis for ℓ p . Define the sequence { x i ∗ k } i ∈ Z ⊂ X ∗ for any i ∈ Z by x i ∗ k = δ e k ∗ if i = k and x i ∗ k = e i ∗ if i ≠ k , where { e i ∗ } i ∈ Z is the dual basic sequence of { e i } i ∈ Z . It is found that { x i ∗ k } i ∈ Z is an ℓ p -frame for ℓ p with bounds 1 and δ . For any x ∈ X d and i ∈ Z , define reconstruction operator S k : X d ↦ X by ( S k ( x ) ) i = δ − 1 x k if i = k and ( S k ( x ) ) i = x i if i ≠ k . Then ( { x i ∗ k } i ∈ Z , S k ) is a Banach frame for ℓ p with respect to ℓ p .

In 1989, Gröchenig extended the concept of classical frames to Banach spaces and called it atomic decomposition. We recall the original definition of atomic decomposition that appears in [14].

Definition 4

Let X be a Banach space and let X d be an associated Banach space of scalar-valued sequences. Let { g i } ⊂ X ∗ be given. If there is a sequence { x i } in X such that for each x ∈ X ,

x = ∑ i ∈ I g i ( x ) x i ,

then ( { g i } , { x i } ) is called the atomic decomposition of X with respect to X d .

In 1999, Casazza et al. introduced the concept of the Schauder frames to the Banach spaces in [16]. The definition of Schauder frames is a natural extension of the dual frame pairs in Hilbert spaces and Schauder bases in Banach spaces.

Definition 5

Let X be a Banach space. The sequence { x i } in X is called a Schauder frame of X , if there exist a BK-space X d and a sequence { g i } in X ∗ such that ( { g i } , { x i } ) is the atomic decomposition of X with respect to X d .

The definition of atomic decomposition expresses the desire to obtain a series expansion of x ∈ X like the frames in Hilbert spaces. On the other hand, the definition of Banach frames opens the way for the possibility of reconstruct formulas. Equation (2.2) can be regarded as a “generalized reconstruction formula,” because it tells us how to restore the original element x ∈ X based on the coefficients { g i ( x ) } . It is well known that there are separable Banach spaces without bases. Casazza et al. proved in [16] that every separable Banach space has a Banach frame. From this point of view, it can be said that the concept of Banach frames is very satisfactory because they always exist.

3 Relay fusion frames in Banach spaces

In this section, we introduce the relay fusion frames in Banach spaces, and we discuss some of their useful properties in this setting.

Definition 6

Let Λ i ∈ ℬ ( X , Y i ) . Suppose that q i is a continuous linear projection on Y i and { p i } is a sequence of continuous linear projections on X and let { v i } be a sequence of weights, i.e., v i > 0 . Write Ω i = p i ( X ) , Δ i = q i ( Y i ) . The sequence { ( Ω i , Δ i , Λ i , v i ) } is called a relay fusion frame for X , or simply R-fusion frame for X , if there exists a solid BK-space X d such that

{ ‖ v i q i Λ i p i ( x ) ‖ } ∈ X d , for all x ∈ X ,
there exist constants 0 < α ≤ β < ∞ such that
(3.1) α ‖ x ‖ ≤ ‖ { ‖ v i q i Λ i p i ( x ) ‖ } ‖ ≤ β ‖ x ‖ .

In this case, we say that { ( Ω i , Δ i , Λ i , v i ) } is an ( α , β ) -R-fusion frame for X . The numbers α , β are called R-fusion frame bounds or simply “bounds.”

In the following remark, we point out the relationship among R-fusion frame and fusion frame and g-frame.

Remark 1

Note that if Y i = X , q i = Λ i = I X for all i ∈ I , then we obtain from Definition 6 the fusion frame { ( Ω i , v i ) } for X . Moreover, assume that p i = I X , q i = I Y i , v i = 1 for each i ∈ I , then Definition 6 covers the g-frames in Banach spaces. The special case, where Y i = C , i ∈ I , gives rise to the X d -frames.

For the existence of R-fusion frames in Banach spaces, we have the following example.

Example 3

Let { ( Ω i , v i ) } be a fusion frame for Banach space X and { ( Δ i , w i ) } be a fusion frame for Banach space Y . Let Λ ∈ ℬ ( X , Y ) be invertible such that T Ω i = Δ i for any i ∈ I . It is easy to check that { ( Ω i , Δ i , Λ i , v i ) } is an R-fusion frame for X and { ( Δ i , Ω i , Λ − 1 , w i ) } is an R-fusion frame for Y . This scenario shows that R-fusion frame model can be carried out in both directions.

Now consider the analysis space of the R-fusion frames in Banach spaces. Define

Σ = ( ⊕ Δ i ) d = { { y i } : y i ∈ Δ i , { ‖ y i ‖ } ∈ X d } .

The following theorem will be essential for the research.

Theorem 1

The analysis space Σ with the norm

‖ { y i } ‖ Σ = ‖ { ‖ y i ‖ } ‖ X d

is a Banach space with coordinate-wise operations.

Proof

For any w = { w i } , z = { z i } in Σ and for every i ∈ I , we have

‖ z i + w i ‖ ≤ ‖ z i ‖ + ‖ w i ‖ .

Since { ‖ w i ‖ } , { ‖ z i ‖ } ∈ X d , and X d is a solid space, we see that { ‖ z i + w i ‖ } ∈ X d and thus z + w ∈ Σ . Further,

‖ z + w ‖ Σ = ‖ { ‖ z i + w i ‖ } ‖ X d ≤ ‖ { ‖ z i ‖ } + { ‖ w i ‖ } ‖ X d ≤ ‖ { ‖ z i ‖ } ‖ X d + ‖ { ‖ w i ‖ } ‖ X d = ‖ z ‖ Σ + ‖ w ‖ Σ .

It is easy to observe that for every λ ∈ C , ‖ λ z ‖ Σ = ∣ λ ∣ ‖ z ‖ Σ . Therefore, Σ is a normed space. Now we show that Σ is complete. Consider the mapping

η : Σ → X d , η ( { y i } ) = { ‖ y i ‖ } , ∀ { y i } ∈ Σ .

For all w = { w i } ∈ Σ , we know ‖ η ( w ) ‖ X d = ‖ w ‖ Σ . Moreover, for any i ∈ I ,

‖ z i ‖ − ‖ w i ‖ ≤ ‖ z i − w i ‖ .

Due to the fact that X d is a solid space, we have

(3.2) ‖ η ( z ) − η ( w ) ‖ X d ≤ ‖ z − w ‖ Σ , ∀ z , w ∈ Σ .

For every i ∈ I , define mapping

P i : Σ → Δ i , P i ( w ) = w i , ∀ w = { w i } ∈ Σ .

Obviously, P i is a projection. By the continuity of coordinate functionals π i : X d → C ,

(3.3) ‖ P i ( w ) ‖ = ‖ w i ‖ = π i ( η ( w ) ) ≤ ‖ π i ‖ ‖ η ( w ) ‖ = ‖ π i ‖ ‖ w ‖ .

It follows that P i is a continuous linear map. Let { w ( n ) } ⊂ Σ be a Cauchy sequence, where w ( n ) = { w i ( n ) } , ∀ n ∈ N . By (3.3), for all i ∈ I , { w i ( n ) } n ⊂ Δ i is a Cauchy sequence. Owing to Δ i is complete, there exists w i ∈ Δ i such that w i ( n ) → w i as n → ∞ . Hence, for each i ∈ I , ‖ w i ( n ) ‖ → ‖ w i ‖ as n → ∞ .

On the other hand, by (3.2), { η ( w ( n ) ) } n is a Cauchy sequence in X d , so there exists λ = { λ i } ∈ X d such that { η ( w ( n ) ) } n converges to λ . Thanks to the continuity of coordinate functionals π i : X d → C , for each i ∈ I , ‖ w i ( n ) ‖ → λ i , n → ∞ . Thus, for all i ∈ I , ‖ w i ‖ = λ i and λ = { ‖ w i ‖ } ∈ X d . Therefore, w = { w i } ∈ Σ .

It can be seen from the aforementioned proof that for every n ∈ N , w ( n ) − w ∈ Σ and { w ( n ) − w } is a Cauchy sequence in Σ . Moreover, for each i ∈ I , w i ( n ) − w i → 0 as n → ∞ . Since ‖ w ( n ) − w ‖ Σ = ‖ η ( w ( n ) − w ) ‖ X d , we have { w ( n ) } → w as n → ∞ . Hence, Σ is complete, which immediately yields the desired conclusion.□

Remark 2

From Theorem 1, it is easy to observe that each Δ i is isomorphic to its copy M i in Σ , where

M i = { { y i δ i j } j ∈ I : y i ∈ Δ i } .

With the help of Theorem 1, we can define the analysis operator for R-fusion frames in Banach spaces.

Definition 7

Let { ( Ω i , Δ i , Λ i , v i ) } be an R-fusion frame for X . The analysis operator for { ( Ω i , Δ i , Λ i , v i ) } is defined by

Θ : X → Σ , Θ ( x ) = { v i q i Λ i p i ( x ) } .

We state some of the important properties of analysis operator Θ :

Theorem 2

Let { ( Ω i , Δ i , Λ i , v i ) } be an R-fusion frame for X. Then the analysis operator Θ is a bounded, injective, linear operator with closed range and Θ − 1 : Θ ( X ) → X is bounded.

Proof

For all x ∈ X , we have

α ‖ x ‖ ≤ ‖ { Θ ( x ) } ‖ ≤ β ‖ x ‖ ,

which implies that Θ is a bounded, injective, linear operator with closed range and Θ − 1 : Θ ( X ) → X is bounded.□

The generalized reconstruction formula (2.2) motivates the following definition.

Definition 8

Let { ( Ω i , Δ i , Λ i , v i ) } be an R-fusion frame for X . If there exists a bounded linear map Γ : Σ → X such that for any x ∈ X ,

Γ ( { v i q i Λ i p i ( x ) } ) = x ,

then we call { ( Ω i , Δ i , Λ i , v i ) , Γ } a Banach R-fusion frame.

According to this definition, we can easily obtain the following result.

Theorem 3

Let { ( Ω i , Δ i , Λ i , v i ) } be an R-fusion frame for X with analysis operator Θ . Then

{ ( Ω i , Δ i , Λ i , v i ) , Θ − 1 }

is a Banach R-fusion frame.

Remark 3

It is easy to verify that if { ( Ω i , Δ i , Λ i , v i ) , Γ } is a Banach R-fusion frame, then Γ Θ = I X and Θ Γ is a projection and therefore Θ ( X ) is a complemented subspace of Σ .

The Schauder frames can be displayed as a special case of the r-fusion frames in a specific sense. We will elaborate on this in the following proposition. We first recall the definition of a biorthogonal system.

Definition 9

Let { x i } ⊂ X and { g i } ⊂ X ∗ , we say that { g i } is biorthogonal to { x i } , or that { ( g i , x i ) } is a biorthogonal system for X , if g j ( x i ) = δ i j for all i , j ∈ I .

Proposition 1

Let { g i } ⊂ X ∗ be an ( α , β ) - X d -frame for X and ( { g i } , { x i } ) be a Schauder frame for X. Suppose T i ∈ ℬ ( X , Y i ) are isometry operators and { g i } and { x i } are biorthogonal. Then

{ ( span { x i } , span { T i x i } , T i , ‖ x i ‖ − 1 ) }

is an ( α , β ) -R-fusion frame for X.

Proof

Assume that { g i } ⊂ X ∗ is an ( α , β ) - X d -frame for X and ( { g i } , { x i } ) is a Schauder frame for X . For every x ∈ X , we have

x = ∑ i ∈ I g i ( x ) x i ,

and

α ‖ x ‖ ≤ ‖ { g i ( x ) } ‖ ≤ β ‖ x ‖ .

Define

p i : X → span { x i } , p i ( x ) = g i ( x ) x i , ∀ x ∈ X .

Since { g i } and { x i } are biorthogonal, p i : X → span { x i } is a continuous linear projection on X . Further,

α ‖ x ‖ ≤ ‖ { ‖ v i q i Λ i p i ( x ) ‖ } ‖ = ‖ { g i ( x ) } ‖ ≤ β ‖ x ‖ ,

where v i = ‖ x i ‖ − 1 . Therefore { ( span { x i } , span { T i x i } , T i , ‖ x i ‖ − 1 ) } is an ( α , β ) -R-fusion frame for X . Hence, the conclusion follows.□

For further research, we need the following result. For convenience, ‖ ⋅ ‖ Z i , d is abbreviated as ‖ ⋅ ‖ i .

Lemma 1

[21] If X d , ( Z i , d , ‖ ⋅ ‖ i ) , i ∈ I are all BK-spaces, then

S d = ( ⊕ Z i , d ) = { { λ i } : λ i = { z i j } j ∈ J i ∈ Z i , d , { ‖ λ i ‖ } ∈ X d }

with the norm ‖ { λ i } ‖ S d = ‖ { ‖ λ i ‖ } ‖ X d is a BK-space. Moreover, if every ( Z i , d , ‖ ⋅ ‖ i ) , i ∈ I is solid, then S d is solid.

We recall the formal definition of g-frames in Banach spaces.

Definition 10

We call a sequence { Λ i ∈ ℬ ( X , Y i ) : i ∈ I } an ( α , β ) -g-frame for X with respect to Y i if there exist a solid BK-space X d and positive constants 0 < α ≤ β < ∞ such that for all x ∈ X , { ‖ Λ i ( x ) ‖ } ∈ X d and

α ‖ x ‖ ≤ ‖ { ‖ Λ i ( x ) ‖ } ‖ ≤ β ‖ x ‖ .

Consider a ( λ i , μ i ) -g-frame { Λ i j ∈ ℬ ( Y i , W i j ) : j ∈ J i } for each Y i with associated BK-space Z i , d , where

0 < λ = inf λ i ≤ sup μ i = μ < ∞ .

In this case we say that { Λ i j ∈ ℬ ( Y i , W i j ) : j ∈ J i } is ( λ , μ ) -bounded for all i ∈ I .

Theorem 4

For each i ∈ I , let S i ∈ ℬ ( X ) , T i ∈ ℬ ( X , Y i ) , R i ∈ ℬ ( Y i ) . Let { Λ i j ∈ ℬ ( Y i , W i j ) : j ∈ J i } be a ( λ i , μ i ) -g-frame for each Y i with associated BK-space Z i , d and suppose that they are ( λ , μ ) -bounded. Then the following conditions are equivalent.

{ R i T i S i ∈ ℬ ( X , Y i ) : i ∈ I } is a g-frame for X with respect to { Y i : i ∈ I } with associated BK-space X d .
{ Λ i j R i T i S i ∈ ℬ ( X , W i j ) : i ∈ I , j ∈ J i } is a g-frame for X with respect to { W i j : i ∈ I , j ∈ J i } with associated BK-space S d .

Proof

Note that for every x ∈ X and each i ∈ I , we have R i T i S i ( x ) ∈ Y i , so { ‖ Λ i j R i T i S i ( x ) ‖ } j ∈ J i ∈ Z i , d and

(3.4) λ ‖ R i T i S i ( x ) ‖ ≤ λ i ‖ R i T i S i ( x ) ‖ ≤ ‖ { Λ i j R i T i S i ( x ) } j ∈ J i ‖ ≤ μ i ‖ R i T i S i ( x ) ‖ ≤ μ ‖ R i T i S i ( x ) ‖ .

(i) ⇒ (ii) Let { R i T i S i ∈ ℬ ( X , Y i ) : i ∈ I } be an ( α , β ) -g-frame for X with respect to { Y i : i ∈ I } . For any x ∈ X , we obtain { ‖ R i T i S i ( x ) ‖ } ∈ X d and

(3.5) α ‖ x ‖ ≤ ‖ { ‖ R i T i S i ( x ) ‖ } ‖ ≤ β ‖ x ‖ .

Since X d is a solid space, by (3.4),

{ ‖ { ‖ Λ i j R i T i S i ( x ) ‖ } j ∈ J i ‖ i } ∈ X d

and

‖ { ‖ { ‖ Λ i j R i T i S i ( x ) ‖ } j ∈ J i ‖ i } ‖ X d ≤ μ ‖ { ‖ R i T i S i ( x ) ‖ } ‖ X d .

By (3.5), we see that

‖ { ‖ { ‖ Λ i j R i T i S i ( x ) ‖ } j ∈ J i ‖ i } ‖ X d ≤ μ β ‖ x ‖ .

Similarly,

‖ { ‖ { ‖ Λ i j R i T i S i ( x ) ‖ } j ∈ J i ‖ i } ‖ X d ≥ α λ ‖ x ‖ ,

which leads to the desired result.

(ii) ⇒ (i) Assume that { Λ i j R i T i S i ∈ ℬ ( X , W i j ) : i ∈ I , j ∈ J i } is an ( α , β ) -g-frame for X with respect to { W i j : i ∈ I , j ∈ J i } with associated BK-space S d . It follows that for any x ∈ X , { { Λ i j R i T i S i ( x ) } j ∈ J i } i ∈ S d and

α ‖ x ‖ ≤ ‖ { ‖ { ‖ Λ i j R i T i S i ( x ) ‖ } j ∈ J i ‖ i } ‖ ≤ β ‖ x ‖ .

By using (3.4), we have { R i T i S i ( x ) } ∈ X d and

‖ { ‖ R i T i S i ( x ) ‖ } ‖ ≤ β λ ‖ x ‖ .

Similarly,

‖ { ‖ R i T i S i ( x ) ‖ } ‖ ≥ α μ ‖ x ‖ ,

which proves the claim.□

We can obtain two equivalent descriptions of the R-fusion frames in Banach spaces as follows by using Theorem 4.

Corollary 1

For every i ∈ I , let T i ∈ ℬ ( X , Y i ) . Suppose that q i is a continuous projection on Y i and { p i } is a family of continuous projection on X and let { v i } be a family of weights, i.e., v i > 0 . Write Ω i = p i ( X ) , Δ i = q i ( Y i ) . If { Λ i j ∈ ℬ ( Y i , W i j ) : j ∈ J i } is a ( λ i , μ i ) -g-frame for each Y i , i ∈ I , and they are ( λ , μ ) -bounded, then the following statements are equivalent.

{ ( Ω i , Δ i , Λ i , v i ) } is an R-fusion frame for X.
{ v i q i T i p i ∈ ℬ ( X , Δ i ) : i ∈ I } is a g-frame for X with respect to { Δ i : i ∈ I } with associated BK-space X d .
{ v i Λ i j q i T i p i ∈ ℬ ( X , W i j ) : i ∈ I , j ∈ J i } is a g-frame for X with respect to { W i j : i ∈ I , j ∈ J i } with associated BK-space S d .

Corollary 2

For every i ∈ I , let T i ∈ ℬ ( X , Y i ) . Suppose that q i is a continuous projection on Y i and { p i } is a family of continuous projection on X and let { v i } be a family of weights, i.e., v i > 0 . Write Ω i = p i ( X ) , Δ i = q i ( Y i ) . If { π i j ∈ ℬ ( Y i , W i j ) : j ∈ J i } is a ( λ i , μ i ) -fusion frame for each Y i , i ∈ I , and they are ( λ , μ ) -bounded, then the following statements are equivalent.

{ ( Ω i , Δ i , Λ i , v i ) } is an R-fusion frame for X.
{ v i q i T i p i ∈ ℬ ( X , Δ i ) : i ∈ I } is a g-frame for X with respect to { Δ i : i ∈ I } with associated BK-space X d .
{ v i π i j q i T i p i ∈ ℬ ( X , W i j ) : i ∈ I , j ∈ J i } is a g-frame for X with respect to { W i j : i ∈ I , j ∈ J i } with associated BK-space S d .

4 Perturbation

The problem studied in this section is the stability of the R-fusion frames in Banach spaces, that is, whether a set of elements close to the R-fusion frame is itself an R-fusion frame. We define ( λ , μ , c ) -perturbation by employing the canonical Paley-Wiener type definition.

Definition 11

Let { ( Ω i , Δ i , Λ i , v i ) } be an R-fusion frame for X , 0 ≤ λ , μ < 1 , c = { c i } ∈ X d . Let Γ i ∈ ℬ ( X , Y i ) . Suppose that τ i is a continuous projection on Y i and { π i } is a family of continuous projection on X and let { u i } be a family of weights, i.e., u i > 0 . Write Φ i = π i ( X ) , Ψ i = τ i ( Y i ) . If for all x ∈ X ,

‖ v i q i Λ i p i ( x ) − u i τ i Γ i π i ( x ) ‖ ≤ λ ‖ v i q i Λ i p i ( x ) ‖ + μ ‖ u i τ i Γ i π i ( x ) ‖ + c i ‖ x ‖ ,

then we say that { ( Φ i , Ψ i , Γ i , u i ) } is a ( λ , μ , c ) -perturbation of { ( Ω i , Δ i , Λ i , v i ) } .

Employing this definition, we derive the following result about robustness of R-fusion frames under small perturbations. Specifically, we show

Theorem 5

Let { ( Ω i , Δ i , Λ i , v i ) } be an ( α , β ) -R-fusion frame for X and let { ( Φ i , Ψ i , Γ i , u i ) } be a ( λ , μ , c ) -perturbation of { ( Ω i , Δ i , Λ i , v i ) } . If 1 − λ 1 + μ α − ‖ c ‖ > 0 , then { ( Φ i , Ψ i , Γ i , u i ) } is an R-fusion frame for X with R-fusion frame bounds

1 − λ 1 + μ α − ‖ c ‖ and ( 1 + μ ) ( 1 + λ ) 1 − μ β + 1 + μ 1 − μ ‖ c ‖ .

Proof

For each x ∈ X , we have

‖ v i q i Λ i p i ( x ) − u i τ i Γ i π i ( x ) ‖ ≤ λ ‖ v i q i Λ i p i ( x ) ‖ + μ ‖ u i τ i Γ i π i ( x ) ‖ + c i ‖ x ‖ .

It follows that

( 1 − μ ) ‖ u i τ i Γ i π i ( x ) ‖ ≤ ( 1 + λ ) λ ‖ v i q i Λ i p i ( x ) ‖ + c i ‖ x ‖ .

Since { ‖ v i q i Λ i p i ( x ) ‖ } ∈ X d and c = { c i } ∈ X d and X d is a solid BK-space, we see that

{ ‖ u i τ i Γ i π i ( x ) ‖ } ∈ X d ,

so we can now conclude that

( 1 − μ ) ‖ { ‖ u i τ i Γ i π i ( x ) ‖ } ‖ ≤ ( 1 + λ ) λ ‖ { ‖ v i q i Λ i p i ( x ) ‖ } ‖ + ‖ c ‖ ‖ x ‖ .

Similarly,

( 1 − λ ) ‖ { ‖ v i q i Λ i p i ( x ) ‖ } ‖ ≤ ( 1 + μ ) λ ‖ { ‖ u i τ i Γ i π i ( x ) ‖ } ‖ + ‖ c ‖ ‖ x ‖ .

Write

A = 1 − λ 1 + μ α − ‖ c ‖ , B = ( 1 + μ ) ( 1 + λ ) 1 − μ β + 1 + μ 1 − μ ‖ c ‖ .

A simple calculation yields

A ‖ x ‖ ≤ ‖ { ‖ u i τ i Γ i π i ( x ) ‖ } ‖ ≤ B ‖ x ‖ .

Therefore, { ( Φ i , Ψ i , Γ i , u i ) } is an R-fusion frame for X with R-fusion frame bounds

1 − λ 1 + μ α − ‖ c ‖ and ( 1 + μ ) ( 1 + λ ) 1 − μ β + 1 + μ 1 − μ ‖ c ‖ .

Hence, the R-fusion frame condition is satisfied with the claimed bounds.□

Theorem 6

Under the setup of Theorem 5, assume further that { ( Ω i , Δ i , Λ i , v i ) , Γ } is a Banach R-fusion frame. If

A > 0 and ‖ Γ ‖ ( β λ + μ B + ‖ c ‖ ) < 1 ,

then there exists a bounded operator ϒ : Σ → X such that { ( Φ i , Ψ i , Γ i , u i ) , ϒ } is a Banach R-fusion frame.

Proof

From the proof of Theorem 5, for any x ∈ X , we have

{ ‖ u i τ i Γ i π i ( x ) ‖ } ∈ X d ,

and therefore, { u i τ i Γ i π i ( x ) } ∈ Σ . Define mapping

S : X → Σ , S ( x ) = { u i τ i Γ i π i ( x ) } .

It is clear that S is a linear operator and ‖ S ‖ ≤ B . Moreover, for every x ∈ X ,

‖ Γ S ( x ) − x ‖ = ‖ Γ ( S − Θ ) ( x ) ‖ ≤ ‖ Γ ‖ ‖ ( S − Θ ) ( x ) ‖ ≤ ‖ Γ ‖ ( β λ + μ B + ‖ c ‖ ) ‖ x ‖ .

Thus,

‖ Γ S − I X ‖ ≤ ‖ Γ ‖ ( β λ + μ B + ‖ c ‖ ) < 1 .

This implies that Γ S ∈ ℬ ( X ) is an invertible operator. Further, let ϒ = ( Γ S ) − 1 Γ . Then ϒ S = I X . Consequently, { ( Φ i , Ψ i , Γ i , u i ) , ϒ } is a Banach R-fusion frame, as desired.□

The following is another version of perturbation of R-fusion frames in Banach spaces.

Theorem 7

Let { ( Ω i , Δ i , Λ i , v i ) } be an ( α , β ) -R-fusion frame for X and let T i ∈ ℬ ( Y i ) be a bounded invertible operator. Suppose that

0 < m = inf 1 ‖ T i − 1 ‖ ≤ sup ‖ T i ‖ = M < ∞ .

Then { ( Ω i , Δ i , T i Λ i , v i ) } is an ( α m , β M ) -R-fusion frame for X.

Proof

Let x ∈ X . For each i ∈ I , we have

m ‖ v i q i Λ i p i ( x ) ‖ ≤ ‖ v i q i T i Λ i p i ( x ) ‖ ≤ M ‖ v i q i Λ i p i ( x ) ‖ .

Since X d is a solid BK-space and { ‖ v i q i Λ i p i ( x ) ‖ } ∈ X d , we obtain { ‖ v i q i T i Λ i p i ( x ) ‖ } ∈ X d and

‖ { ‖ v i q i T i Λ i p i ( x ) ‖ } ‖ X d ≤ M ‖ { ‖ v i q i Λ i p i ( x ) ‖ } ‖ X d .

Similarly, we have

m ‖ { ‖ v i q i Λ i p i ( x ) ‖ } ‖ X d ≤ ‖ { ‖ v i q i T i Λ i p i ( x ) ‖ } ‖ X d .

Hence, for every x ∈ X ,

α m ‖ x ‖ ≤ ‖ { ‖ v i q i T i Λ i p i ( x ) ‖ } ‖ X d ≤ β M ‖ x ‖ .

Therefore, our assertion is tenable.□

Let us end this section with the following result, which provides explicit estimates for the R-fusion frame bounds of an R-fusion frame of the form { ( T Ω i , Δ i , Λ i , v i ) } .

Theorem 8

Let { ( Ω i , Δ i , Λ i , v i ) } be an ( α , β ) -R-fusion frame for X and let T ∈ ℬ ( X ) be a bounded invertible operator. Then { ( T Ω i , Δ i , Λ i , v i ) } is an R-fusion frame for X with bounds

α ‖ T ‖ ‖ T − 1 ‖ and ‖ T ‖ ‖ T − 1 ‖ β .

Proof

Since { ( Ω i , Δ i , Λ i , v i ) } is an ( α , β ) -R-fusion frame for X , each Ω i is complemented in X , and there exists a solid BK-space X d such that R-fusion frame inequality (3.1) holds for any x ∈ X . Due to the fact that T ∈ ℬ ( X ) is invertible, we obtain T Ω i is complemented and T p i T − 1 is a continuous linear projection on T Ω i for each i ∈ I . Moreover, for every i ∈ I and x ∈ X , we have

1 ‖ T ‖ 1 ‖ T − 1 ‖ α ‖ x ‖ ≤ ‖ { ‖ v i q i Λ i T p i T − 1 ( x ) ‖ } ‖ X d ≤ ‖ T ‖ ‖ T − 1 ‖ β ‖ x ‖ ,

which completes the proof.□

Acknowledgments

The authors wish to thank the anonymous reviewers for their valuable comments and suggestions that have improved the presentation of this article.

Funding information: This work was supported by the National Natural Science Foundation of China (Nos. 11671201 and 11771379).
Author contributions: All authors contributed to each part of this work equally and read and approved the final manuscript.
Conflict of interest: The authors declare that they have no competing interest.
Ethical approval: Not applicable.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2022-07-31

Revised: 2022-12-09

Accepted: 2023-01-21

Published Online: 2023-02-14

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2022-0554

Keywords for this article

relay fusion frame; Banach relay fusion frame; perturbation; Banach space

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