Characterization and Construction of K-Fusion Frames and Their Duals in Hilbert Spaces

TL;DR

This paper introduces K-fusion frames in Hilbert spaces, providing methods for their identification, dual construction, and analyzing their robustness, with applications in sampling theory.

Contribution

It defines K-fusion frames, offers new approaches for their characterization and dual construction, and studies their stability under perturbations.

Findings

K-fusion frames can be characterized via multiple approaches.

Explicit methods for constructing duals of K-fusion frames are provided.

K-fusion frames exhibit robustness under certain perturbations.

Abstract

K-frames, a new generalization of frames, were recently considered by L. Gavruta in connection with atomic systems and some problems arising in sampling theory. Also, fusion frames are an important generalization of frames, applied in a variety of applications. In the present paper, we introduce the notion of K-fusion frames in Hilbert spaces and obtain several approaches for identifying of $K$-fusion frames. The main purpose is to reconstruct the elements from the range of the bounded operator $K$ on a Hilbert space H by using a family of closed subspaces in H. This work will be useful in some problems in sampling theory which are processed by fusion frames. For this end, we present some descriptions for duality of K-fusion frames and also resolution of the operator K to provide simple and concrete constructions of duals of K-fusion frames. Finally, we survey the robustness of K-fusion frames under some perturbations.

Equations

$$A\Vert K^{\ast }f{\Vert }^2\le \sum _{i\in I}|⟨f,f_i⟩{|}^2\le B\Vert f{\Vert }^2,(f\in \mathcal{H}).$$

$$Kf=\sum _{i\in I}⟨f,g_i⟩f_i,(f\in \mathcal{H}).$$

$$A\Vert f{\Vert }^2\le \sum _{i\in I}{\omega }_i^2\Vert {\pi }_{W_i}f{\Vert }^2\le B\Vert f{\Vert }^2,(f\in \mathcal{H}),$$

$$\sum _{i\in I}\oplus W_i=\left\{\{f_i{\}}_{i\in I}:f_i\in W_i,\sum _{i\in I}\Vert f_i{\Vert }^2<\infty \right\},$$

$$T_W\left(\{f_i{\}}_{i\in I}\right)=\sum _{i\in I}{\omega }_i{f}_i,\left(\{f_i{\}}_{i\in I}\in \sum _{i\in I}\oplus W_i\right).$$

$$T_W^{\ast }(f)={\left\{{\omega }_i{\pi }_{W_i}(f)\right\}}_{i\in I},(f\in \mathcal{H}).$$

$$f=\sum _{i\in I}{\omega }_i{\nu }_i{\pi }_{V_i}{S}_W^{-1}{\pi }_{W_i}f,(f\in \mathcal{H}).$$

$$A\Vert K^{\ast }f{\Vert }^2\le \sum _{i\in I}{\omega }_i^2\Vert {\pi }_{W_i}f{\Vert }^2\le B\Vert f{\Vert }^2,(f\in \mathcal{H}).$$

$$W_1=span\{e_1,e_2\},W_2=span\{e_3\}.$$

$$K{e}_1=e_1,K{e}_2=e_1,K{e}_3=e_2.$$

$$A⟨K{K}^{\ast }f,f⟩\le \sum _{i\in I}{\omega }_i^2\Vert {\pi }_{W_i}f{\Vert }^2=⟨S_{W}f,f⟩,$$

$${\pi }_V{T}^{\ast }={\pi }_V{T}^{\ast }{\pi }_{\overline{TV}}.$$

$$\sum _{i\in I}{\omega }_i^2\Vert {\pi }_{T{W}_i}f{\Vert }^2$$

$$\sum _{i\in I}{\omega }_i^2\Vert {\pi }_{K^{†}{W}_i}f{\Vert }^2$$

$$A\Vert f{\Vert }^2$$

$$A\Vert K^{\ast }{Q}^{\ast }f{\Vert }^2$$

$$\frac{A}{{\lambda }^2}\Vert Q^{\ast }f{\Vert }^2\le \sum _{i\in I}{\omega }_i^2\Vert {\pi }_{W_i}f{\Vert }^2,$$

$$\sum _{i\in I}{\omega }_i^2\Vert {\pi }_{K{W}_i}f{\Vert }^2$$

$$A\Vert K^{\ast }f{\Vert }^2$$

$$T_W{Q}^{\ast }{T}_V^{\ast }=K.$$

$$\Vert Kf{\Vert }^4$$

$$C\ge B^{-1}\Vert Q{\Vert }^{-2}\text{and}D\ge A^{-1}\Vert Q{\Vert }^{-2},$$

$$T_{W}X=K.$$

$$f\in (X_i^{\ast }{W}_i{)}^{\perp }=(R(X_i^{\ast }){)}^{\perp }=N(X_i),(i\in I),$$

$$\Vert \Gamma \{{\pi }_{X_i^{\ast }{W}_i}f{\}}_{i\in I}{\Vert }^2$$

$$T_W{Q}^{\ast }{T}_{\hat{W}}^{\ast }=T_{W}X=K,$$

$$K{e}_1=e_1+e_2,K{e}_2=e_3,K{e}_3=0,$$

$$W_1=\text{span}\{e_1+e_2,e_3\},W_2=\text{span}\{e_3\},W_3=\text{span}\{e_1+e_2\},$$

$$Xf=\left\{\left(\frac{a}{2},\frac{a}{2},\frac{b}{2}\right),\left(0,0,\frac{b}{2}\right),\left(\frac{a}{2},\frac{a}{2},0\right)\right\}$$

Full text

Key words: Fusion frames; $K$-fusion frames; $K$-duals; resolution of bounded operators.

1. Introduction and preliminaries

Frame theory presents efficient algorithms for a wide range of applications [2, 4, 6, 7, 8]. In most of those applications, we deal with dual frames to reconstruct the modified data and compare it with the original data. In contrast to frames, a new approach so called atomic decomposition for a closed subspace $\mathcal{H}_{0}$ of a Hilbert space $\mathcal{H}$ introduced by Feichtinger et al. in [16] with frame-like properties. However, the sequences in atomic decompositions do not necessarily belong to $\mathcal{H}_{0}$, this striking property is valuable especially in sampling theory [26, 28]. Then $K$-frames were introduced to study atomic systems with respect to a bounded operator $K\in B(\mathcal{H})$ [18]. Indeed, $K$-frames are equivalent with atomic systems for the operator $K$ and help us to reconstruct elements from the range of a bounded linear operator $K$ in a separable Hilbert space. More precisely, let $\mathcal{H}$ be a separable Hilbert space and $I$ a countable index set, a sequence $F:=\{f_{i}\}_{i\in I}\subseteq\mathcal{H}$ is called a $K$-frame for $\mathcal{H}$, if there exist constants $A,B>0$ such that

[TABLE]

Clearly, if $K=I_{\mathcal{H}}$, then $F$ is an ordinary frame and so $K$-frames arise as a generalization of the ordinary frames [8, 13, 18]. The constants $A$ and $B$ in $(\ref{001})$ are called the lower and the upper bounds of $F$, respectively. Similar to ordinary frames the synthesis operator can be defined as $T_{F}:l^{2}\rightarrow\mathcal{H}$; $T_{F}(\{c_{i}\}_{i\in I})=\sum_{i\in I}c_{i}f_{i}$. It is a bounded operator and its adjoint which is called the analysis operator given by $T_{F}^{*}(f)=\{\langle f,f_{i}\rangle\}_{i\in I}$, and the frame operator is given by $S_{F}:\mathcal{H}\rightarrow\mathcal{H}$; $S_{F}f=T_{F}T_{F}^{*}f=\sum_{i\in I}\langle f,f_{i}\rangle f_{i}$. Unlike ordinary frames, the frame operator of a K-frame is not invertible in general. However, if $K$ has close range then $S_{F}$ from $R(K)$ onto $S_{F}(R(K))$ is an invertible operator [29].

The authors in [1] introduced the notion of duality for $K$-frames and presented some methods for construction and characterization of $K$-frames and their duals. Indeed, a Bessel sequence $\{g_{i}\}_{i\in I}\subseteq\mathcal{H}$ is called a $K$-dual of $\{f_{i}\}_{i\in I}$ if

[TABLE]

For further information in $K$-frame theory we refer the reader to [1, 16, 18, 29]. The following result is useful for the proof of our main results.

Theorem 1.1** (Douglas [15]).**

Let $L_{1}\in B(\mathcal{H}_{1},\mathcal{H})$ and $L_{2}\in B(\mathcal{H}_{2},\mathcal{H})$ be bounded linear mappings on given Hilbert spaces. Then the following assertions are equivalent:

• (i)

$R(L_{1})\subseteq R(L_{2})$;

• (ii)

$L_{1}L_{1}^{*}\leq\lambda^{2}L_{2}L_{2}^{*}$,  for some $\lambda>0$;

• (iii)

There exists a bounded linear mapping $X\in L(\mathcal{H},\mathcal{H}_{2})$, such that $L_{1}=L_{2}X$.

Moreover, if (i), (ii) and (iii) are valid, then there exists a unique operator $X$ so that

• (a)

$\|X\|^{2}=\inf\{\alpha>0,L_{1}L_{1}^{*}\leq\alpha L_{2}L_{2}^{*}\}$;

• (b)

$N(L_{1})=N(X)$;

• (c)

$R(X)\subset\overline{R(L_{2}^{*})}$.

Fusion frame theory is a fundamental mathematical theory introduced in [9] to model sensor networks perfectly. Although, recent studies shows that fusion frames provide effective frameworks not only for modeling of sensor networks but also for signal and image processing, sampling theory, filter banks and a variety of applications that cannot be modeled by discrete frames [11, 22, 25]. In the following, we review basic definitions and results of fusion frames.

Let $\{W_{i}\}_{i\in I}$ be a family of closed subspaces of $\mathcal{H}$ and $\{\omega_{i}\}_{i\in I}$ a family of weights, i.e. $\omega_{i}>0$, $i\in I$. Then $\{(W_{i},\omega_{i})\}_{i\in I}$ is called a fusion frame for $\mathcal{H}$ if there exist the constants $0<A\leq B<\infty$ such that

[TABLE]

where $\pi_{W_{i}}$ denotes the orthogonal projection from Hilbert space $\mathcal{H}$ onto a closed subspace $W_{i}$. The constants $A$ and $B$ are called the fusion frame bounds. If $\omega_{i}=1$, for all $i$, $W$ is called uniform fusion frame and we denote it by $\{W_{i}\}_{i\in I}$. Also, if we only have the upper bound in (1.3) we call $\{(W_{i},\omega_{i})\}_{i\in I}$ a Bessel fusion sequence. Recall that for each sequence $\{W_{i}\}_{i\in I}$ of closed subspaces in $\mathcal{H}$, the space

[TABLE]

with the inner product $\left\langle\{f_{i}\}_{i\in I},\{g_{i}\}_{i\in I}\right\rangle=\sum_{i\in I}\langle f_{i},g_{i}\rangle$ is a Hilbert space.

For a Bessel fusion sequence $W:=\{(W_{i},\omega_{i})\}_{i\in I}$ of $\mathcal{H}$, the synthesis operator $T_{W}:\sum_{i\in I}\oplus W_{i}\rightarrow\mathcal{H}$ is defined by

[TABLE]

Its adjoint operator $T_{W}^{*}:\mathcal{H}\rightarrow\sum_{i\in I}\oplus W_{i}$, which is called the analysis operator, is given by

[TABLE]

and the fusion frame operator $S_{W}:\mathcal{H}\rightarrow\mathcal{H}$ is defined by $S_{W}f=\sum_{i\in I}\omega_{i}^{2}\pi_{W_{i}}f$, which is a bounded, invertible and positive operator [9].

There are some approaches towards dual fusion frames, the first definition was presented by P. G$\breve{\textrm{a}}$vruţa in [17]. A Bessel fusion sequence $\{(V_{i},\nu_{i})\}_{i\in I}$ is called a dual fusion frame of $\{(W_{i},\omega_{i})\}_{i\in I}$ if

[TABLE]

The family $\{(S_{W}^{-1}W_{i},\omega_{i})\}_{i\in I}$, which is also a fusion frame, is called the canonical dual of $\{(W_{i},\omega_{i})\}_{i\in I}$. A general approach to dual fusion frames can be found in [20, 21].

Throughout this paper, we suppose $\mathcal{H}$ is a separable Hilbert space, $K^{{\dagger}}$ the pseudo inverse of operator $K$, $I$ a countable index set and $I_{\mathcal{H}}$ is the identity operator on $\mathcal{H}$. For two Hilbert spaces $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ we denote by $B(\mathcal{H}_{1},\mathcal{H}_{2})$ the collection of all bounded linear operators between $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$, and we abbreviate $B(\mathcal{H},\mathcal{H})$ by $B(\mathcal{H})$. Also we denote the range of $K\in B(\mathcal{H})$ by $R(K)$, the null space of $K$ by $N(K)$ and the orthogonal projection of $\mathcal{H}$ onto a closed subspace $V\subseteq\mathcal{H}$ by $\pi_{V}$.

The paper is organized as follows. In Section 2, we describe the notion of $K$-fusion frames and present several methods for identifying and constructing of $K$-fusion frames. Section 3 deals with the duality of $K$-fusion frames, in this section, we introduce the notion of dual for $K$-fusion frames and then we show that in case $K=I_{\mathcal{H}}$ this definition coincides with the concept of dual fusion frames, however there are several essentially differences, which we will discuss. Also, we present some characterizations for duals of $K$-fusion frames. Section 4 is devoted to introduce the concept of resolution of a bounded linear operator $K\in B(\mathcal{H})$. By applying this notion we obtain more reconstructions from the elements of $R(K)$. Finally, we survey the robustness of $K$-fusion frames and their duals under some perturbations, in Section 5.

2. $K$-fusion frames

In this section, we introduce the notion of $K$-fusion frames in Hilbert spaces and discuss on some their properties. In particular, we present some approaches for identifying and constructing of $K$-fusion frames. Let us start our consideration with formal definition of $K$-fusion frames.

Definition 2.1**.**

Let $\{W_{i}\}_{i\in I}$ be a family of closed subspaces of $\mathcal{H}$ and $\{\omega_{i}\}_{i\in I}$ a family of weights, i.e. $\omega_{i}>0$, $i\in I$. We call $W=\{(W_{i},\omega_{i})\}_{i\in I}$ a $K$-fusion frame for $\mathcal{H}$, if there exist positive constants $0<A,B<\infty$ such that

[TABLE]

The constants $A$ and $B$ in (2.1) are called lower and upper bounds of $W$, respectively. We call $W$ a minimal $K$-fusion frame, whenever $W_{i}\cap\overline{span}_{j\in I,j\neq i}W_{j}=\{0\}$ and it is called exact, if for every $j\in I$ the sequence $\{(W_{i},\omega_{i})\}_{i\in I,i\neq j}$ is not a $K$- fusion frame for $\mathcal{H}$. Obviously, a $K$-fusion frame is a Bessel fusion sequence and so the synthesis operator, the analysis operator and the frame operator of $W$ are defined similar to fusion frames, however for a $K$-fusion frame, the synthesis operator is not onto and the frame operator is not invertible, in general. Furthermore, there are several other differences between fusion frames and $K$-fusion frames. Indeed, the closed linear span of $W_{i}$’s which contains $R(K)$ by Theorem 1.1, is not equal to $\mathcal{H}$. Also, the following example shows that, unlike fusion frames, a minimal $K$-fusion frame is not necessarily required to be exact. Take $\mathcal{H}=\mathbb{R}^{4}$ with the orthonormal basis $\{e_{i}\}_{i=1}^{4}$ and

[TABLE]

Define $K\in B(\mathcal{H})$ as

[TABLE]

Then $W=\{(W_{i},1)\}_{i=1}^{2}$ is a minimal $K$-fusion frame with the bounds $1/2$ and $1$. However, it is not exact since $\{(W_{1},1)\}$ is also a $K$-fusion frame with the same bounds. In this paper, we will recognize more differences and similarities of $K$-fusion frames with fusion frames.

Proposition 2.2**.**

Suppose that $\{(W_{i},\omega_{i})\}_{i\in I}$ is a Bessel fusion sequence and $K\in B(\mathcal{H})$ is a closed range operator. The following statements are equivalent.

The sequence $W=\{(W_{i},\omega_{i})\}_{i\in I}$ is a $K$-fusion frame for $\mathcal{H}$.

There exists a positive number $A$ such that $S_{W}\geq AKK^{*}$.

Proof.

Since $W$ is a Bessel fusion sequence, so it is a $K$-fusion frame for $\mathcal{H}$ if and only if there exists $A>0$ such that

[TABLE]

for every $f\in\mathcal{H}$, or equivalently $S_{W}\geq AKK^{*}$. ∎

Notice that, if $F$ is a Bessel sequence, unlike frames and fusion frames, invertibility of the frame operator $S_{F}:R(K)\rightarrow S_{F}(R(K))$ does not imply that $F$ is a $K$-frame. For a simple counterexample, let $K$ be the orthogonal projection onto the subspace generated by $(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})$. Then $F=\{(1,0)\}$ is not a $K$-frame for $\mathbb{C}^{2}$, however the operator $S_{F}:R(K)\rightarrow S_{F}(R(K))$ is invertible.

The following lemmas are necessary for our results.

Lemma 2.3**.**

[11]** Let $V$ be a closed subspace of $\mathcal{H}$ and $T$ be a bounded operator on $\mathcal{H}$. Then

[TABLE]

The next lemma was shown for fusion frames in [27], although we prove it by a simple method.

Lemma 2.4**.**

Let $T\in B(\mathcal{H}_{1},\mathcal{H}_{2})$ be an invertible operator and $W=\{(W_{i},\omega_{i})\}_{i\in I}$ be a Bessel fusion sequence of $\mathcal{H}_{1}$. Then $\{(TW_{i},\omega_{i})\}_{i\in I}$ is a Bessel fusion sequence of $\mathcal{H}_{2}$.

Proof.

By applying Lemma 2.3 and the fact that $T$ is invertible, we obtain

[TABLE]

for each $f\in\mathcal{H}_{2}$, as required. ∎

Theorem 2.5**.**

Let $K\in B(\mathcal{H})$ be a closed range operator and $W=\{(W_{i},\omega_{i})\}_{i\in I}$ a $K$-fusion frame for $\mathcal{H}$ with bounds $A$ and $B$, respectively. Then

• (i)

If $\{\pi_{R(K)}W_{i},\omega_{i})\}_{i\in I}$ is a Bessel fusion sequence, then $\{(K^{{\dagger}}W_{i},\omega_{i})\}_{i\in I}$ is a fusion frame for $R(K^{*})$.

• (ii)

If $\{\pi_{S_{W}(R(K))}W_{i},\omega_{i})\}_{i\in I}$ is a Bessel fusion sequence, then $\{(S_{W}^{-1}\pi_{S_{W}(R(K))}W_{i},\omega_{i})\}_{i\in I}$ is a fusion frame for $R(K)$.

• (iii)

If $Q\in B(\mathcal{H})$ is an invertible operator, then $\{(QW_{i},\omega_{i})\}_{i\in I}$ is a $QK$-fusion frame for $\mathcal{H}$.

• (iv)

If $Q\in B(\mathcal{H})$ is an invertible operator and $KQ=QK$. Then $QW$ is also a $K$-fusion frame.

• (v)

If $Q\in B(\mathcal{H})$ such that $R(Q)\subseteq R(K)$, then $W$ is also a $Q$-fusion frame.

Proof.

To show $\{(K^{{\dagger}}W_{i},\omega_{i})\}_{i\in I}$ is a fusion frame for $R(K^{*})$, let $f\in R(K^{*})$. Then, we can write

[TABLE]

Hence by assumption $\{(K^{{\dagger}}W_{i},\omega_{i})\}_{i\in I}$ is a Bessel fusion sequence. On the other hand, there exists $A>0$ such that

[TABLE]

Therefore, $(i)$ holds. Since the sequence $\{\omega_{i}\pi_{W_{i}}e_{j}\}_{i\in I,j\in J}$ is a $K$-frame for $\mathcal{H}$ the sequence $\{(K^{*})^{{\dagger}}K^{*}S_{W}^{-1}\pi_{S_{W}(R(K))}\omega_{i}\pi_{W_{i}}e_{j}\}_{i\in I,j\in J}$ is a frame for $R(K)$, by Corollary 1 in [1]. Hence, there exists $A>0$ such that

[TABLE]

for each $f\in R(K)$. Now, since $S_{W}^{-1}:S_{W}(R(K))\rightarrow R(K)$ is an invertible operator so $\{(S_{W}^{-1}\pi_{S_{W}(R(K))}W_{i},\omega_{i})\}_{i\in I}$ is a Bessel fusion sequence for $R(K)$ by Lemma 2.4, this follows $(ii)$. To show $(iii)$, suppose that $Q\in B(\mathcal{H})$ is an invertible operator, then $\{(QW_{i},\omega_{i})\}_{i\in I}$ is a Bessel sequence, by Theorem 2.4 in [17]. Moreover,

[TABLE]

for every $f\in\mathcal{H}$. Thus, $\{(QW_{i},\omega_{i})\}_{i\in I}$ is a $QK$-fusion frame. The part $(iv)$ is obtained by $(iii)$. Finally, for $(v)$, we have $R(Q)\subseteq R(K)$ so there exists $\lambda>0$ such that $QQ^{*}\leq\lambda^{2}KK^{*}$ by Proposition 1.1. This follows that

[TABLE]

Therefore $W$ is a $Q$-fusion frame for $\mathcal{H}$. ∎

Notice that, the condition in Theorem 2.5 (ii) is established in many statuses. For example, if for all $i\in I$ either $W_{i}\subseteq S_{W}(R(K))$ or $\pi_{S_{W}(R(K))}W_{i}\subseteq W_{i}$. Also, Theorem 2.5 $(v)$ is a generalization of Proposition 3.3 in [24]. In the end of this section, we present the second approach for constructing of $K$-fusion frames.

Theorem 2.6**.**

Let $K$ be a closed range operator and $W=\{(W_{i},\omega_{i})\}_{i\in I}$ a fusion frame for $R(K^{*})$. Then $\{(KW_{i},\omega_{i})\}_{i\in I}$ is a $K$-fusion frame for $\mathcal{H}$.

Proof.

It is not difficult to see that every Bessel fusion sequence for a closed subspace of $\mathcal{H}$ is also a Bessel fusion sequence for $\mathcal{H}$. Suppose $B$ is an upper bound for $W$ as a Bessel fusion sequence for $\mathcal{H}$. Also, let $f\in\mathcal{H}$ we can write $f=g+h$ which $g\in R(K)$ and $h\in(R(K))^{\perp}$. Thus

[TABLE]

Hence $\{(KW_{i},\omega_{i})\}_{i\in I}$ is a Bessel fusion sequence. Moreover, there exists $A>0$ such that

[TABLE]

This follows the result. ∎

By applying Theorem 2.6 the following result immediately is obtained.

Corollary 2.7**.**

Let $K$ be a closed range operator and $\{(W_{i},\omega_{i})\}_{i\in I}$ a fusion frame for $\mathcal{H}$. Then $\{(K(W_{i}\cap R(K^{*})),\omega_{i})\}_{i\in I}$ is a $K$-fusion frame for $\mathcal{H}$.

3. Duality of $K$-fusion frames

In this section, we present some descriptions for duality of $K$-fusion frames. Then, we try to characterize and identify duals of $K$-fusion frames. Our approach to define the duality of $K$-fusion frames is a generalization of the idea in [21].

Definition 3.1**.**

Let $W=\{(W_{i},\omega_{i})\}_{i\in I}$ be a $K$-fusion frame, a Bessel fusion sequence $V=\{(V_{i},\upsilon_{i})\}_{i\in I}$ is called a $QK$-dual of $W$ if there exists a bounded linear operator $Q:\sum_{i\in I}\oplus W_{i}\rightarrow\sum_{i\in I}\oplus V_{i}$ such that

[TABLE]

Every $QK$-dual of $W$ is a $K^{*}$-fusion frame. More precisely, if $V=\{(V_{i},\upsilon_{i})\}_{i\in I}$ is a $QK$-dual of $W$, we can write

[TABLE]

for every $f\in\mathcal{H}$, where $B$ is an upper bound of $W$. Moreover, if $C$ and $D$ are the optimal bounds of $V$, respectively. Then

[TABLE]

in which $A$ and $B$ are the optimal bounds of $W$, respectively.

Remark 3.2*.*

Consider a $K$-fusion frame $W=\{(W_{i},\omega_{i})\}_{i\in I}$ for $\mathcal{H}$. Applying the Douglas’ theorem [15] there exists an operator $X\in B(\mathcal{H},\sum_{i\in I}\oplus W_{i})$ such that

[TABLE]

We denote the $i$-th component of $Xf$ by $X_{i}f=(Xf)_{i}$ and clearly $X_{i}\in B(\mathcal{H},W_{i})$.

In the next theorem, we show that by these operators one may construct some $QK$-duals for $W$.

Theorem 3.3**.**

Let $W=\{(W_{i},\omega_{i})\}_{i\in I}$ be a $K$-fusion frame and $X$ be an operator as in $(\ref{DougX})$. If $\widehat{W}=\{X_{i}^{*}W_{i}\}_{i\in I}$ is a Bessel fusion sequence, then it is a $QK$-dual for $W$.

Proof.

Define the mapping $\Gamma:R(T_{\widehat{W}}^{*})\rightarrow\sum_{i\in I}\oplus W_{i}$ so that $\Gamma T_{\widehat{W}}^{*}f=Xf$. Then $\Gamma$ is well-defined and bounded. Indeed, for every $f\in\mathcal{H}$ if $\Gamma T_{\widehat{W}}^{*}f=\{\pi_{X_{i}^{*}W_{i}}f\}_{i\in I}=0$ we imply that

[TABLE]

i.e., $Xf=0$. Moreover,

[TABLE]

Hence, $\Gamma$ can be uniquely extended to $\overline{R(T_{\widehat{W}}^{*})}$. Also, we take $\Gamma=0$ on $R(T_{\widehat{W}}^{*})^{\perp}$ and let $Q=\Gamma^{*}$. This implies that $Q^{*}\in B(\sum_{i\in I}\oplus X_{i}^{*}W_{i},\sum_{i\in I}\oplus W_{i})$ and

[TABLE]

as required. ∎

Example 3.4**.**

Consider $\mathcal{H}=\mathbb{R}^{3}$ and define $K\in B(\mathcal{H})$ as

[TABLE]

where $\{e_{i}\}_{i=1}^{3}$ is the standard orthonormal basis of $\mathcal{H}$. Also let

[TABLE]

and $\omega_{i}=1$, for all $1\leq i\leq 3$. Then $W=\{(W_{i},\omega_{i})\}_{i=1}^{3}$ is a $K$-fusion frame with bounds $1$ and $2$, respectively. Now, define the operator $X:\mathbb{R}^{3}\rightarrow\sum_{i\in 1}^{3}\oplus W_{i}$ as

[TABLE]

for every $f=(a,b,c)\in\mathbb{R}^{3}$. One can easily see that $T_{W}X=K$ and

[TABLE]

Hence, $\{X_{i}^{*}W_{i}\}_{i\in I}$ is a $QK$-dual for $W$. Moreover, $\|X\|=1$, $\|K^{*}f\|^{2}=(a+b)^{2}+c^{2}$ and $\|T_{W}^{*}f\|^{2}=(a+b)^{2}+2c^{2}$ and so

[TABLE]

Also, $N(X)=N(K)$ and $R(X)\subseteq\overline{R(T_{W}^{*})}$. This shows that the operator $X$ is the unique operator, which satisfies all items in Douglas’ theorem.

Notice that in Theorem 3.3, $\widehat{W}$ is not necessarily Bessel fusion sequence. In fact, a simple computation shows that $X^{*}\{g_{i}\}_{i\in I}=\sum_{i\in I}X_{i}^{*}g_{i}$, for all $\{g_{i}\}_{i\in I}\in\sum_{i\in I}\oplus W_{i}$. So for every $K$-fusion frame such that $W_{i}\perp W_{j}$, for all $i\neq j$, we obtain

[TABLE]

Now, let $\{e_{i}\}_{i\in I}$ be an orthonormal basis of $\mathcal{H}$ and $W_{i}=span\{e_{i}\}$, for all $i\in I$. Clearly $\{W_{i}\}_{i\in I}$ is an orthonormal fusion basis and also a $K$-fusion frame for $\mathcal{H}$. Define

[TABLE]

Then the mapping $K^{*}$ can be extended to a bounded and surjective linear operator on $\mathcal{H}$, i.e., $K^{*}\in B(\mathcal{H})$. Moreover,

[TABLE]

Thus, for $f=e_{1}$

[TABLE]

i.e. $\{X_{i}^{*}W_{i}\}_{i=1}^{\infty}=\{K^{*}W_{i}\}_{i=1}^{\infty}$ is not a Bessel fusion sequence.

Using the Douglas’ theorem, the equation $T_{W}X=K$ has a unique solution as $X_{w}$ such that

[TABLE]

It is worth to note that, in case $K=I_{\mathcal{H}}$ we obtain $X_{w}=T_{W}S_{W}^{-1}$ and so the $QK$-dual $\{X_{i}^{*}W_{i}\}_{i\in I}$ of $W$ is exactly $\{S_{W}^{-1}W_{i}\}$. By these considerations, we can obtain optimal bounds of a $K$-fusion frame. Let $W=\{(W_{i},\omega_{i})\}_{i\in I}$ be $K$-fusion frame with optimal bounds $A$ and $B$, respectively. Then the upper bound is obtained directly by definition as $B=\|S_{W}\|$. Also

[TABLE]

As a considerable result, we get the optimal lower bound of fusion frames.

Corollary 3.5**.**

Let $W=\{(W_{i},\omega_{i})\}_{i\in I}$ be a fusion frame with the optimal lower bound $A$. Then

[TABLE]

Recall that, a bounded operator $Q:\sum_{i\in I}\oplus W_{i}\rightarrow\sum_{i\in I}\oplus V_{i}$ is component preserving [21], whenever

[TABLE]

where

[TABLE]

If the operator $Q$ in (3.1) is component preserving then $V$ is called $QK$-component preserving dual of $W$. By a similar argument with [21] we obtain the following characterization of $QK$-component preserving duals of a $K$-fusion frame, so we avoid the burden of proof.

Theorem 3.6**.**

Let $W=\{(W_{i},\omega_{i})\}_{i\in I}$ be a $K$-fusion frame such that $\omega_{i}>\delta>0$, for some $\delta>0$ and $i\in I$. Then a Bessel fusion sequence $V=\{(V_{i},\upsilon_{i})\}_{i\in I}$ is a $QK$-component preserving dual of $W$ if and only if $V_{i}=\Psi p_{i}\sum_{j\in I}\oplus W_{j}$, in which $\Psi\in B(\sum_{i\in I}\oplus W_{i},\mathcal{H})$ such that $\Psi T_{W}^{*}=K^{*}$.

3.1. $K$-Duals

In the squel, we present the other approach to reconstruct the elements of $R(K)$. To this end, we generalize duality introduced by G$\breve{\textrm{a}}$vruţa in [17]. This approach gives us an explicit form for dual of $K$-fusion frames, which is coincident with the canonical dual of fusion frames in case $K=I_{\mathcal{H}}$. Moreover, we obtain several methods for constructing and characterization of duals of $K$-fusion frames. Let $W=\{(W_{i},\omega_{i})\}_{i\in I}$ be a $K$-fusion frame, we can write

[TABLE]

Hence, we obtain the following definition, which is also a special status of (3.1) by taking

[TABLE]

Definition 3.7**.**

Let $W=\{(W_{i},\omega_{i})\}_{i\in I}$ be a $K$-fusion frame. A Bessel fusion sequence $\{(V_{i},\upsilon_{i})\}_{i\in I}$ is called a $K$-dual of $W$ if

[TABLE]

Remark 3.8*.*

• (a)

Let $K=I_{\mathcal{H}}$ in (3.4), we easily see that $V$ is a dual of $W$ in the notion of [17].

• (b)

If $\widetilde{W}:=\{(K^{*}S_{W}^{-1}\pi_{S_{W}(R(K))}W_{i},\omega_{i})\}_{i\in I}$ is a Bessel fusion sequence, then it is a $K$-dual for $W$ and in this case we call it the canonical $K$-dual of $W$.

• (c)

The sequence $\widetilde{W}$ is not a Bessel fusion sequence, necessarily. In the following, we illustrate this fact.

Example 3.9**.**

Let $\mathcal{H}=l^{2}$ with the standard orthonormal basis $\{e_{n}\}_{n=1}^{\infty}$. Define

[TABLE]

Then $K\in B(\mathcal{H})$ and $K^{*}:\mathcal{H}\rightarrow\mathcal{H}$ is given by

[TABLE]

Now, take $W_{i}=\overline{span}\{e_{i}\}$, for all $i\neq 2,4$, $W_{2}=W_{4}=\overline{span}\{e_{2}+e_{4}\}$ and $\omega_{i}=1$, for all $i$. Then $\{W_{i}\}_{i=1}^{\infty}$ is a $K$-fusion frame. More precisely, for every $f=\{a_{i}\}_{i=1}^{\infty}\in\mathcal{H}$ we have

[TABLE]

Furthermore,

[TABLE]

and

[TABLE]

Therefore, a direct calculation shows that

[TABLE]

for all $1\leq m<\infty$, i.e., $\left\{K^{*}S_{W}^{-1}\pi_{S_{W}(R(K))}W_{i}\right\}_{i=1}^{\infty}$ is not a Bessel fusion sequence.

It is worth noticing that, when $\{(\pi_{S_{W}(R(K))}W_{i},\omega_{i})\}_{i\in I}$ is a Bessel fusion sequence with a Bessel bound $B$ then $\widetilde{W}=\{(K^{*}S_{W}^{-1}\pi_{S_{W}(R(K))}W_{i},\omega_{i})\}_{i\in I}$ is also a Bessel fusion sequence with the Bessel bound $B\|K\|^{2}\|K^{\dagger}\|^{2}\|S_{W}\|^{2}\|S_{W}^{-1}\|^{2}$. To show this, assume that $f\in\mathcal{H}$ and $f=g+h$, where $g\in R(K^{*})$ and $h\in(R(K^{*}))^{\perp}$, then

[TABLE]

where the last inequality is obtained by Lemma 2.4.

Now, we are going to present a simple method for constructing of $K$-duals by the canonical $K$-dual. For this, let $W=\{(W_{i},\omega_{i})\}_{i\in I}$ be a $K$-fusion frame with the canonical $K$-dual $\widetilde{W}$ such that $K^{*}S_{W}^{-1}\pi_{S_{W}(R(K))}W_{j}\neq\mathcal{H}$, for some $j\in I$. This implies that $(K^{*}S_{W}^{-1}\pi_{S_{W}(R(K))}W_{j})^{\perp}\neq\{0\}$. Take

[TABLE]

where $U_{j}$ is a closed subspace of $\left(K^{*}S_{W}^{-1}\pi_{S_{W}(R(K))}W_{j}\right)^{\perp}$, and for all $i\neq j$ consider $V_{i}=K^{*}S_{W}^{-1}\pi_{S_{W}(R(K))}W_{i}$. Then, $V=\{(V_{i},\omega_{i})\}_{i\in I}$ is a Bessel fusion sequence and clearly it is a $K$-dual of $\{(W_{i},\omega_{i})\}_{i\in I}$ different from the canonical $K$-dual. More precisely, for every $f\in\mathcal{H}$

[TABLE]

Now, let us turn to the example.

Example 3.10**.**

Suppose $\mathcal{H}$, $K$ and $W$ are as in Example 3.4. Then we have

[TABLE]

Therefore,

[TABLE]

Now, a straightforward calculation shows that $S_{W}^{-1}\pi_{S_{W}(R(K))}W_{i}=W_{i}$, for every $1\leq i\leq 3$. Hence, the canonical $K$-dual $\widetilde{W}=\{(K^{*}S_{W}^{-1}\pi_{S_{W}(R(K))}W_{i},\omega_{i})\}_{i\in I}$ is obtained as the following

$K^{*}S_{W}^{-1}\pi_{S_{W}(R(K))}W_{1}=\textit{span}\{e_{1},e_{2}\}$,

$K^{*}S_{W}^{-1}\pi_{S_{W}(R(K))}W_{2}=\textit{span}\{e_{2}\}$,

$K^{*}S_{W}^{-1}\pi_{S_{W}(R(K))}W_{3}=\textit{span}\{e_{1}\}$.

Also, consider

[TABLE]

Then, $\{(V_{i},1)\}_{i\in I}$ is a $K$-dual of $W$ different from the canonical $K$-dual.

It is worth to note that, in the above example the canonical $K$-dual is exactly the unique $QK$-dual of Example 3.4. This comes from the fact that, in this $K$-fusion frame $S_{W}(R(K))\subset R(K)$. More general, we have the following result.

Theorem 3.11**.**

Let $K$ be a closed range operator and $W=\{(W_{i},\omega_{i})\}_{i\in I}$ a $K$-fusion frame for $\mathcal{H}$. Then, $\widetilde{W}=\{(X_{w}^{*})_{i}W_{i}\}_{i\in I}$ if and only if $S_{W}(S_{W}(R(K)))\subseteq R(K)$.

Proof.

First, suppose $S_{W}(S_{W}(R(K)))\subseteq R(K)$. Obviously, the equation

[TABLE]

has a solution as $M:=T_{W}^{*}(S_{W}^{-1})^{*}K$. Applying the assumption we obtain $T_{W}M=K$, i.e., $M$ satisfies $(\ref{DougX})$. Also, for every $f\in N(M)$ we obtain

[TABLE]

Hence, $N(M)=N(K)$ and clearly $R(M)\subseteq\overline{R(T_{W}^{*})}$. Thus, $M=X_{w}$ by Theorem 1.1. Also, $M_{i}=\pi_{W_{i}}(S_{W}^{-1})^{*}K$ so

[TABLE]

or equivalently, $\widetilde{W}=\{(X_{w}^{*})_{i}W_{i}\}_{i\in I}$. Conversely, let $\widetilde{W}=\{(X_{w}^{*})_{i}W_{i}\}_{i\in I}$. Then,

[TABLE]

for all $\{f_{i}\}_{i\in I}\in\sum_{i\in I}\oplus W_{i}$. Therefore, $X_{w}=T_{W}^{*}(S_{W}^{-1})^{*}K$, and the operator $T_{W}^{*}(S_{W}^{-1})^{*}K$ satisfies

[TABLE]

i.e., $S_{W}(S_{W}(R(K)))\subseteq R(K)$, as required. ∎

In the sequel, we characterizes all $K$-duals of minimal $K$-fusion frames, under some condition. For this, we need to a simple lemma, which prove it for convenience.

Lemma 3.12**.**

Let $K$ be a closed range operator and $F=\{f_{i}\}_{i\in I}$ be a $K$-frame for $\mathcal{H}$. Then $\{\pi_{R(K)}f_{i}\}_{i\in I}$ is also a $K$-frame for $\mathcal{H}$ with $K$-dual $\{K^{*}S_{F}^{-1}\pi_{S_{F}(R(K))}f_{i}\}_{i\in I}$.

Proof.

Since the operator $S_{F}:R(K)\rightarrow S_{F}(R(K))$ is invertible. This follows that $\{K^{*}S_{F}^{-1}\pi_{S(R(K))}f_{i}\}_{i\in I}$ is a Bessel sequence. Hence

[TABLE]

for all $f\in\mathcal{H}$. So the result follows. ∎

Theorem 3.13**.**

Let $K$ be a closed range operator and $W=\{(W_{i},\omega_{i})\}_{i\in I}$ a minimal $K$-fusion frame for $\mathcal{H}$ with the canonical $K$-dual $\widetilde{W}$. Also, assume that $\overline{span}\{W_{i}\}_{i\in I}\cap R(K)^{\perp}=\{0\}$. Then a Bessel fusion sequence $V=\{(V_{i},\omega_{i})\}_{i\in I}$ is a $K$-dual of $W$ if and only if

[TABLE]

Proof.

Suppose that $\{e_{i,j}\}_{j\in J_{i}}$ is an orthonormal basis of $W_{i}$, for all $i\in I$. Then we can easily see that the sequence $F=\{\omega_{i}e_{i,j}\}_{i\in I,j\in J_{i}}$ is a $K$-minimal frame for $\mathcal{H}$ and $S_{F}=S_{W}$. Hence, $\{\omega_{i}\pi_{R(K)}e_{i,j}\}_{i\in I,j\in J_{i}}$ is a $K$-minimal frame for $\mathcal{H}$, so it has a unique $K$-dual by Theorem 6 in [1] and this dual is $\{K^{*}S_{W}^{-1}\pi_{S_{W}(R(K))}\omega_{i}e_{i,j}\}_{i\in I,j\in J_{i}}$, by Lemma 3.12. Now, let $V$ be a $K$-dual of $W$. Then

[TABLE]

for every $f\in\mathcal{H}$. This shows that the sequence $\{\pi_{V_{i}}K^{*}S_{W}^{-1}\pi_{S_{W}(R(K))}\omega_{i}e_{i,j}\}_{i\in I,j\in J_{i}}$ is a $K$-dual of $F$. Hence,

[TABLE]

Thus

[TABLE]

i.e., $K^{*}S_{W}^{-1}\pi_{S_{W}(R(K))}W_{i}\subseteq V_{i}$, for all $i\in I$. Conversely, if a Bessel fusion sequence $V$ satisfies $K^{*}S_{W}^{-1}\pi_{S_{W}(R(K))}W_{i}\subseteq V_{i}$, for all $i\in I$. Then

[TABLE]

This shows that $V$ is a $K$-dual of $W$. ∎

As a consequence we regain the following result, which was proved in [3] for fusion frames.

Corollary 3.14**.**

Let $W=\{(W_{i},\omega_{i})\}_{i\in I}$ be a minimal fusion frame for $\mathcal{H}$. Then a Bessel fusion sequence $V=\{(V_{i},\omega_{i})\}_{i\in I}$ is a dual of $W$ if and only if $S_{W}^{-1}W_{i}\subseteq V_{i}$, for all $i\in I$.

Remark 3.15*.*

Consider a $K$-fusion frame $W=\{(W_{i},\omega_{i})\}_{i\in I}$ for $\mathcal{H}$ and let $F_{i}=\{f_{i,j}\}_{j\in J_{i}}$ be a frame for $W_{i}$, for each $i\in I$ with frame bounds $A_{i}$ and $B_{i}$, respectively such that $0<A=\inf_{i\in I}A_{i}\leq B=\sup_{i\in I}B_{i}<\infty$. Then the sequences $\{f_{i,j}\}_{j\in J_{i}}$ are called local frames of $W$ and $\{(W_{i},\omega_{i},\{f_{i,j}\}_{j\in J_{i}})\}_{i\in I}$ is called a $K$-fusion frame system. Also, if $\{\widetilde{f}_{i,j}\}_{j\in J_{i}}$ is a dual for $F_{i}$ in $W_{i}$, we call $\{\widetilde{f}_{i,j}\}_{j\in J_{i}}$ local dual frames.

The following results describe the duality of $K$-fusion frames with respect to local frames.

Theorem 3.16**.**

Let $W=\{(W_{i},\omega_{i})\}_{i\in I}$ be a $K$-fusion frame and $V=\{(V_{i},\upsilon_{i})\}_{i\in I}$ be a Bessel fusion sequence. Also, let $\{g_{i,j}\}_{j\in J_{i}}$ be a local frame for $V_{i}$ with bounds $A_{i}$ and $B_{i}$, for all $i\in I$ and the canonical local dual frame $\{\widetilde{g}_{i,j}\}_{j\in J_{i}}$. Then $V$ is a $K$-dual of $W$ if and only if the sequence $G=\{\upsilon_{i}g_{i,j}\}_{i\in I,j\in J_{i}}$ is a $K$-dual of $F=\{\omega_{i}\pi_{R(K)}\pi_{W_{i}}(S_{W}^{-1})^{*}K\widetilde{g}_{i,j}\}_{i\in I,j\in J_{i}}$.

Proof.

We first show that $F$ and $G$ are Bessel sequences for $\mathcal{H}$.

[TABLE]

for every $f\in\mathcal{H}$, where $D$ is an upper bound for $W$ and $A=\inf_{i\in I}A_{i}$. Moreover, we have

[TABLE]

where $B=\sup_{i\in I}B_{i}$. On the other hand,

[TABLE]

Hence, $V=\{(V_{i},\upsilon_{i})\}_{i\in I}$ is a $K$-dual of $W$ if and only if $G$ is a $K$-dual of $F$.∎

By a similar argument to the proof of Theorem 3.16 one may prove the next theorem.

Theorem 3.17**.**

Let $W=\{(W_{i},\omega_{i})\}_{i\in I}$ be a $K$-fusion frame with bounds $A$ and $B$, respectively. A Bessel fusion sequence $V=\{(V_{i},\upsilon_{i})\}_{i\in I}$ is a $K$-dual of $W$ if and only if $G=\{\upsilon_{i}\pi_{V_{i}}e_{j}\}_{i\in I,j\in J}$ is a $K$-dual of $F=\{\omega_{i}\pi_{R(K)}\pi_{W_{i}}(S_{W}^{-1})^{*}Ke_{j}\}_{i\in I,j\in J}$, where $\{e_{j}\}_{j\in J}$ is an orthonormal basis of $\mathcal{H}$.

The following result shows that for every local frame of a $K$-fusion frame we can construct some $K$-frames with associated $K$-duals.

Proposition 3.18**.**

Let $W=\{(W_{i},\omega_{i})\}_{i\in I}$ be a $K$-fusion frame for $\mathcal{H}$ and $\{f_{i,j}\}_{j\in J_{i}}$ be a local frame for $W_{i}$ with the local dual frame $\{\widetilde{f}_{i,j}\}_{j\in J_{i}}$, for all $i\in I$. Then $\{\omega_{i}f_{i,j}\}_{i\in I,j\in J_{i}}$ is a $K$-frame for $\mathcal{H}$ with $K$-dual $G=\{X_{i}^{*}\widetilde{f}_{i,j}\}_{i\in I,j\in J_{i}}$, where the operator $X$ is as in $(\ref{DougX})$.

Proof.

First, note that $G=\{X_{i}^{*}\widetilde{f}_{i,j}\}_{i\in I,j\in J_{i}}$ is a Bessel sequence. In fact,

[TABLE]

for every $f\in\mathcal{H}$ where $B$ is given by Remark 3.15. Also, similar to Theorem 3.2 of [9], we can see that $F=\{\omega_{i}f_{i,j}\}_{i\in I,j\in J_{i}}$ is a $K$-frame for $\mathcal{H}$. Moreover,

[TABLE]

Hence $G$ is a $K^{*}$-frame and also a $K$-dual of $F$. ∎

4. Resolution of bounded linear operators

The concept of resolution of the identity has been considered in [9, 23]. In this section, we introduce the notion of resolution of a bounded linear operator $K\in B(\mathcal{H})$, which lead to more reconstructions from the elements of $R(K)$.

Let $K\in B(\mathcal{H})$ and $\{\theta_{i}\}_{i\in I}$ be a family of bounded linear operators on $\mathcal{H}$, we say $\{\theta_{i}\}_{i\in I}$ is an $l^{2}$-resolution of $K$ with respect to a family of weights $\{\omega_{i}\}_{i\in I}$ for $\mathcal{H}$ whenever there exists a positive constant $B$ such that

• (i)

$Kf=\sum_{i\in I}\omega_{i}^{2}\theta_{i}f$,

• (ii)

$\sum_{i\in I}\omega_{i}^{2}\|\theta_{i}f\|^{2}\leq B\|f\|^{2}$,

for every $f\in\mathcal{H}$. If $\theta:=\{\theta_{i}\}_{i\in I}$ only satisfies $(i)$ we say $\theta$ is a resolution of the operator $K$.

Remark 4.1*.*

• (1)

One can easily shows that for every $l^{2}$-resolution $\{\theta_{i}\}_{i\in I}$ of an operator $K\in B(\mathcal{H})$ there exists $A>0$ such that

[TABLE]

• (2)

Let $W=\{(W_{i},\omega_{i})\}_{i\in I}$ be a $K$-fusion frame for $\mathcal{H}$. Then

• a)

There exists a bounded operator $X\in B(\mathcal{H},\sum_{i\in I}\oplus W_{i})$, such that $K=T_{W}X$ by Theorem 1.1. Hence, the operators $\theta_{i}:\mathcal{H}\rightarrow W_{i}$ given by $\theta_{i}f=X_{i}f$, where $X_{i}f$ is the $i$-th component of $Xf$, constitute an $l^{2}$-resolution of $K$ with respect to the family of weights $\{\sqrt{\omega_{i}}\}_{i\in I}$.

• b)

Define $\theta_{i}\in B(\mathcal{H})$ by $\theta_{i}=\pi_{R(K)}\pi_{W_{i}}(S_{W}^{-1})^{*}K$, for all $i\in I$. Then $\{\theta_{i}\}_{i\in I}$ is an $l^{2}$-resolution of $K$ with respect to $\{\omega_{i}\}_{i\in I}$.

• c)

Suppose $\theta_{i}\in B(\mathcal{H})$ is given by $\theta_{i}=S_{W}^{-1}\pi_{S_{W}(R(K))}\pi_{W_{i}}K$, for all $i\in I$. Then $\{\theta_{i}\}_{i\in I}$ is an $l^{2}$-resolution of $K$ with respect to $\{\omega_{i}\}_{i\in I}$.

As we observed, by using $K$-fusion frames we will obtain many resolutions of the operator $K$. In the following proposition, we show that by an $l^{2}$-resolution of the operator $K$ one may construct a $K$-fusion frame.

Proposition 4.2**.**

Let $\{\theta_{i}\}_{i\in I}$ be an $l^{2}$-resolution of $K$ with respect to $\{\omega_{i}\}_{i\in I}$ for $\mathcal{H}$, such that $W=\{(\overline{R(\theta_{i})},\omega_{i})\}_{i\in I}$ constitute a Bessel fusion sequence. Then $W$ is a $K$-fusion frame for $\mathcal{H}$.

Proof.

By using assumption, it is enough to show the existence of a lower bound for $W$,

[TABLE]

for every $f\in\mathcal{H}$, in which $B$ is an upper bound of $\{\theta_{i}\}_{i\in I}$. Thus, the result follows. ∎

The next theorem shows that the $l^{2}$-resolution constructed by $X_{w}$ has minimum $l^{2}$-norm between all $l^{2}$-resolutions of the operator $K$, where $X_{w}$ is as in $(\ref{XW})$.

Theorem 4.3**.**

Let $W=\{(W_{i},\omega_{i})\}_{i\in I}$ be a $K$-fusion frame for $\mathcal{H}$ and the operators $\theta_{i}:\mathcal{H}\rightarrow W_{i}$ constitute an $l^{2}$-resolution of $K$. Then

[TABLE]

Furthermore,

[TABLE]

Proof.

Suppose that $\{\theta_{i}\}_{i\in I}$ is an $l^{2}$-resolution of $K$. Define $\theta:\mathcal{H}\rightarrow\sum_{i\in I}\oplus W_{i}$ by $\theta f=\{\theta_{i}f\}_{i\in I}$. Then $\theta$ is a bounded linear operator and $T_{W}\theta f=\sum_{i\in I}\theta_{i}f=Kf$. Moreover,

[TABLE]

which follows that

[TABLE]

as required. On the other hand

[TABLE]

This completes the proof. ∎

In the case $K=I_{\mathcal{H}}$, the above theorem reduces to a result in [23]. As a result of Theorem 4.3, we can obtain the pseudo-inverse of the bounded operator $\pi_{R(K)}T_{W}$.

Corollary 4.4**.**

Suppose that $\{(W_{i},\omega_{i})\}_{i\in I}$ is a $K$-fusion frame for $\mathcal{H}$. Then the pseudo-inverse operator $(\pi_{R(K)}T_{W})^{\dagger}:R(K)\rightarrow\sum_{i\in I}\oplus W_{i}$ is given by

[TABLE]

Proof.

For a $K$-fusion frame $W=\{(W_{i},\omega_{i})\}_{i\in I}$ we can easily survey that $T_{W}^{*}|_{R(K)}$ is a one to one operator, so the operator $\pi_{R(K)}T_{W}:\sum_{i\in I}\oplus W_{i}\rightarrow R(K)$ is onto. Let $f\in\mathcal{H}$, by Corollary 1.1 in [5], the equation $Kf=\pi_{R(K)}T_{W}\{f_{i}\}_{i\in I}$ has a unique solution of minimal norm and this solution is $(\pi_{R(K)}T_{W})^{\dagger}Kf$. On the other hand,

[TABLE]

Thus, the result follows by Theorem 4.3. ∎

5. Perturbation of $K$-fusion frames

In fusion frame theory, the elements of underlying Hilbert spaces are distributed to a family of closed subspaces. These elements can be reconstructed by dual fusion frames such as (1.4). In real applications, under these transmissions usually a part of the data vectors change or reshape, in the other words, the various disturbances and perturbations affect on the information. In this respect, stability of fusion frames and dual fusion frames under perturbations has a key role in practice. In this section, we study the robustness of $K$-fusion frames and their $K$-duals under some perturbations.

Theorem 5.1**.**

Let $W=\{(W_{i},\omega_{i})\}_{i\in I}$ be a $K$-fusion frame for $\mathcal{H}$ with bounds $A$ and $B$, respectively. Also, let $Z=\{(Z_{i},z_{i})\}_{i\in I}$ be a $(\lambda_{1},\lambda_{2},\varepsilon)$-perturbation of $W$ for some $0<\lambda_{1},\lambda_{2}<1$ and $\varepsilon>0$, i.e.,

[TABLE]

for all $i\in I$ and $f\in\mathcal{H}$ such that

[TABLE]

Then $Z$ is a $K$-fusion frame for $\mathcal{H}$.

Proof.

We first show the existence of a Bessel bound for $Z$. Let $f\in\mathcal{H}$,

[TABLE]

for every $f\in\mathcal{H}$. Using (5.1), shows that

[TABLE]

Now, it is sufficient to find a lower bound. In fact,

[TABLE]

Therefore

[TABLE]

This completes the proof. ∎

Take $\omega_{i}=z_{i}$ for all $i\in I$ and $K=I_{\mathcal{H}}$, then Theorem 5.1 reduces in Proposition 5.2 of [11]. In the next theorem we show that under small perturbations, $K$-duals of a $K$-fusion frame turn to the approximate $K$-dual for perturbed $K$-fusion frame. Let $W=\{(W_{i},\omega_{i})\}_{i\in I}$ be a $K$-fusion frame for $\mathcal{H}$. A Bessel fusion sequence $V=\{(V_{i},\nu_{i})\}_{i\in I}$ is called an approximate $K$-dual of $W$ whenever $\|K-T_{W}\phi_{vw}T^{*}_{V}\ \|<1$. Approximate duals was first introduced in [14] for discrete frames and are important tools for reconstruction algorithms.

Theorem 5.2**.**

Let $W=\{(W_{i},\omega_{i})\}_{i\in I}$ be a $K$-fusion frame for $\mathcal{H}$ with bounds $A$ and $B$, respectively. Also, let $\{Z_{i}\}_{i\in I}$ be a family of closed subspaces in $\mathcal{H}$.

[TABLE]

for some $\varepsilon>0$. Then,

• (i)

If $0<\varepsilon<\sqrt{A}$, then $Z=\{(Z_{i},\omega_{i})\}_{i\in I}$ is a $K$-fusion frame for $\mathcal{H}$ with the bounds $(\sqrt{A}-\varepsilon)$ and $(\sqrt{B}+\varepsilon\|K\|)$, respectively.

• (ii)

*Every $K$-dual $V=\{(V_{i},\omega_{i})\}_{i\in I}$ of $W$ is an approximate $K$-dual of $Z$, for $\varepsilon>0$ sufficiently small, . *

Proof.

For every $f\in\mathcal{H}$, we can write

[TABLE]

This shows $(i)$. Moreover, let $V=\{(V_{i},\omega_{i})\}_{i\in I}$ be a $K$-dual of $W$. Then

[TABLE]

Now, suppose that

[TABLE]

Therefore,

[TABLE]

for every $f\in\mathcal{H}$. This implies that $\|K-T_{Z}\phi_{vz}T^{*}_{V}\pi_{R(K)}\|<1$, as required. ∎

Example 5.3**.**

Suppose that $\mathcal{H}$, $K$ and $W=\{(W_{i},\omega_{i})\}_{i=1}^{3}$ are as in Example 3.10. Also, let

[TABLE]

Then, the Bessel fusion sequence $Z=\{(Z_{i},1)\}_{i=1}^{3}$ satisfies

[TABLE]

for every $f=(a,b,c)\in\mathcal{H}$, i.e., $Z$ is an $\varepsilon$-perturbation of $W$ with $\varepsilon=1/2$. Thus $Z$ is a $K$-fusion frame by Theorem 5.2 (i). Now, a direct calculation shows that

[TABLE]

and consequently

[TABLE]

Hence, $\|K^{*}(S_{Z}^{-1}-S_{W}^{-1})\|=1/6$ and $\|(S_{Z}^{-1})^{*}K\|=1/3$. Also, we have $A=1$, $B=2$ and $\|K\|=1$. Thus

[TABLE]

Therefore, every $K$-dual $V=\{(V_{i},1)\}_{i=1}^{3}$ of $W$ is an approximate $K$-dual of $Z$, moreover $\|K-T_{Z}\phi_{vz}T^{*}_{V}\pi_{R(K)}\|<1/4$.