K-g-Fusion Frames on Cartesian Products of Two Hilbert C^∗-Modules

1. Introduction

The concept of frames, first introduced by Duffin and Schaeffer [3], provides stable yet redundant representations of vectors in Hilbert spaces. Since its inception, frame theory has become a fundamental tool with wide-ranging applications in harmonic analysis, wavelet theory, signal processing, sampling theory, and operator theory see [1].

Several extensions of frame theory have been proposed to address increasingly sophisticated settings, including g-frames [11], and fusion frames [2], among others [6,9,10]. Each of these generalizations enhances the flexibility of frame representations while preserving their fundamental stability properties. In this context, the notion of K-g-fusion frames, which unifies the features of K-frames, g-frames, and fusion frames, offers a powerful framework for studying operator-related decompositions in Hilbert spaces and beyond.

A natural direction of research has been the extension of frame theory to Hilbert $C^{*}$-modules, initiated by Frank and Larson [4]. In contrast to Hilbert spaces, Hilbert $C^{*}$-modules present significant challenges, arising from the absence of projections onto arbitrary closed submodules and the presence of a $C^{*}$-algebra-valued inner product. Despite these difficulties, frame concepts have been successfully adapted, leading to a variety of results in this setting, see [12,13,14].

The aim of this paper is to advance the theory of K-g-fusion frames on Cartesian products of Hilbert $C^{*}$-modules. Such products naturally emerge in operator algebras, module decompositions, and block-matrix methods, and hence provide a rich framework for our study.

The paper is organized as follows. Section 2 reviews the fundamental notions of Hilbert $C^{*}$-modules and adjointable operators, and introduces the concept of K-g-fusion frames together with their operator-theoretic features. Section 3 contains the main result concerning Cartesian products of K-g-fusion frames. In Section 4, we establish perturbation results, while the final section is devoted to concluding remarks and illustrative examples.

2. Preliminaries

We briefly recall the basic definitions and facts about Hilbert $C^{*}$-modules needed in the sequel. Standard references are [5,7].

Definition 1.  

Let $\mathcal{A}$ be a unital $C^{*}$-algebra. A left Hilbert $C^{*}$-module over $\mathcal{A}$ is a left $\mathcal{A}$-module $\mathcal{H}$ equipped with a map

$$\langle ·,·\rangle :\mathcal{H}\times \mathcal{H}\to \mathcal{A}$$

called the$\mathcal{A}$-valued inner product, satisfying:

• $\langle ax+y,z\rangle =a\langle x,z\rangle +\langle y,z\rangle$ for all $a\in \mathcal{A}$, $x,y,z\in \mathcal{H}$;
• $\langle x,y\rangle ={\langle y,x\rangle }^{*}$ for all $x,y\in \mathcal{H}$;
• $\langle x,x\rangle \ge 0$ in $\mathcal{A}$, and $\langle x,x\rangle =0\iff x=0$.

The associated norm is defined by $\parallel x\parallel :={\parallel \langle x,x\rangle \parallel }^{1/2}$, and completeness with respect to this norm is assumed.

For Hilbert $\mathcal{A}$-modules $\mathcal{H},\mathcal{K}$, we denote by ${\mathrm{End}}_{\mathcal{A}}^{*}(\mathcal{H},\mathcal{K})$ the set of adjointable operators from $\mathcal{H}$ into $\mathcal{K}$, i.e., those operators $T:\mathcal{H}\to \mathcal{K}$ for which there exists an adjoin $T^{*}:\mathcal{K}\to \mathcal{H}$ satisfying

$$\langle Tx,y\rangle =\langle x,T^{*}y\rangle \mathrm{for}\mathrm{all}x\in \mathcal{H},y\in \mathcal{K}.$$

If $\mathcal{K}=\mathcal{H}$, then we simply write ${\mathrm{End}}_{\mathcal{A}}^{*}\left(\mathcal{H}\right)$ instead of ${\mathrm{End}}_{\mathcal{A}}^{*}(\mathcal{H},\mathcal{H}).$ An operator $T\in {\mathrm{End}}_{\mathcal{A}}^{*}\left(\mathcal{H}\right)$ is called positive, written $T\ge 0$, if

$$\langle Tx,x\rangle \ge 0\mathrm{for}\mathrm{all}x\in \mathcal{H}.$$

The partial order on self-adjoint operators is determined by this cone.

For a closed submodule $W\subset \mathcal{H}$, an orthogonal projection $P_W\in {\mathrm{End}}_{\mathcal{A}}^{*}\left(\mathcal{H}\right)$ satisfies

$$P_W^2=P_W=P_W^{*}\mathrm{and}Ran\left(P_W\right)=W.$$

Unlike the Hilbert space case, not every closed submodule is complemented in $\mathcal{H}$.

In this work we restrict attention to orthogonally complemented submodules.

In what follows, all sums indexed by a countable set I are assumed to converge in norm in $\mathcal{A}$ whenever convergence is asserted.

These preliminaries allow us to introduce K-g-fusion frames in Hilbert $C^{*}$-modules in the next.

Given two Hilbert $\mathcal{A}$-modules ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$, their external direct sum (or product) is defined as

$${\mathcal{H}}_1\oplus {\mathcal{H}}_2:=\{(x_1,x_2):x_1\in {\mathcal{H}}_1,x_2\in {\mathcal{H}}_2\},$$

with the natural left $\mathcal{A}$-module action

$$a·(x_1,x_2):=(a{x}_1,a{x}_2),a\in \mathcal{A},$$

and $\mathcal{A}$-valued inner product

$$\langle (x_1,x_2),(y_1,y_2)\rangle :={\langle x_1,y_1\rangle }_{{\mathcal{H}}_1}+{\langle x_2,y_2\rangle }_{{\mathcal{H}}_2}.$$

With this structure, ${\mathcal{H}}_1\oplus {\mathcal{H}}_2$ is a Hilbert $\mathcal{A}$-module.

Moreover, if $V\subset {\mathcal{H}}_1$ and $W\subset {\mathcal{H}}_2$ are orthogonally complemented submodules, then their direct sum $V\oplus W$ is an orthogonally complemented submodule of ${\mathcal{H}}_1\oplus {\mathcal{H}}_2$, with projection operator

$$P_{V\oplus W}=P_V\oplus P_W.$$

This observation will be essential in constructing product families of frames and proving stability under perturbations.

Definition 2.  

Let $\mathcal{H}$ and $\mathcal{K}$ be countably generated Hilbert $\mathcal{A}$-modules. Suppose that:

• ${\left\{v_i\right\}}_{i\in I}$ is a family of positive invertible elements from the center of $\mathcal{A}$;
• ${\left\{W_i\right\}}_{i\in I}$ is a family of orthogonally complemented closed submodules of $\mathcal{H}$;
• ${\left\{{\mathcal{H}}_i\right\}}_{i\in I}$ is a family of closed submodules of $\mathcal{K}$;
• for each $i\in I$, ${\Lambda }_i\in {End}_{\mathcal{A}}^{*}(\mathcal{H},{\mathcal{H}}_i)$;
• $K\in {End}_{\mathcal{A}}^{*}\left(\mathcal{H}\right)$.

We say that $\Lambda ={\left(W_i,{\Lambda }_i,v_i\right)}_{i\in I}$ is aK-g-fusion framefor $\mathcal{H}$ with respect to ${\left\{{\mathcal{H}}_i\right\}}_{i\in I}$ if there exist scalars $0<A\le B<\infty$ such that

$$A\langle K^{*}f,K^{*}f\rangle \le \sum _{i\in I}{v}_i^2\langle {\Lambda }_i{P}_{W_i}f,{\Lambda }_i{P}_{W_i}f\rangle \le B\langle f,f\rangle ,\forall f\in \mathcal{H}.$$

The constants A and B are called thelowerandupper boundsof the K-g-fusion frame. In addition:

• If the inequalities hold with $K=I_{\mathcal{H}}$, then Λ is ag-fusion frame, i.e.

  $$A\langle f,f\rangle \le \sum _{i\in I}{v}_i^2\langle {\Lambda }_i{P}_{W_i}f,{\Lambda }_i{P}_{W_i}f\rangle \le B\langle f,f\rangle ,\forall f\in \mathcal{H}.$$
• If, in addition, $K=I_{\mathcal{H}}$ and ${\Lambda }_i=P_{W_i}$ for all $i\in I$, then Λ reduces to afusion framefor $\mathcal{H}$.

Now, for a K-g-fusion frame $\Lambda ={(W_i,{\Lambda }_i,v_i)}_{i\in I}$ of $\mathcal{H}$ with respect to ${\left\{{\mathcal{H}}_i\right\}}_{i\in I}$,

• The analysis operator

  $$T_{\Lambda }^{*}:\mathcal{H}⟶{\ell }^2\left({\left\{{\mathcal{H}}_i\right\}}_{i\in I}\right),$$

  is defined by

  $$T_{\Lambda }^{*}f={\left(v_i{\Lambda }_i{P}_{W_i}f\right)}_{i\in I},f\in \mathcal{H}.$$
• The synthesis operator

  $$T_{\Lambda }:{\ell }^2\left({\left\{{\mathcal{H}}_i\right\}}_{i\in I}\right)⟶\mathcal{H},$$

  is the adjoint of $T_{\Lambda }^{*}$ and is given by

  $$T_{\Lambda }\left({\left(f_i\right)}_{i\in I}\right)=\sum _{i\in I}{v}_i{P}_{W_i}{\Lambda }_i^{*}{f}_i,{\left(f_i\right)}_{i\in I}\in {\ell }^2\left({\left\{{\mathcal{H}}_i\right\}}_{i\in I}\right).$$
• The frame operator

  $$S_{\Lambda }:\mathcal{H}⟶\mathcal{H}$$

  is defined by

  $$S_{\Lambda }f=T_{\Lambda }{T}_{\Lambda }^{*}f=\sum _{i\in I}{v}_i^2{P}_{W_i}{\Lambda }_i^{*}{\Lambda }_i{P}_{W_i}f,f\in \mathcal{H}.$$

3. Product K-g-Fusion Frames and Main Theorem

Let $\mathcal{A}$ be a unital $C^{*}$-algebra and let ${\mathcal{H}}_1,{\mathcal{H}}_2$ be Hilbert $\mathcal{A}$-modules. For each $i\in I$ let $W_i\subset {\mathcal{H}}_1$ and $V_i\subset {\mathcal{H}}_2$ be orthogonally complemented closed submodules with projections $P_{W_i}\in {\mathrm{End}}_{\mathcal{A}}^{*}\left({\mathcal{H}}_1\right)$ and $P_{V_i}\in {\mathrm{End}}_{\mathcal{A}}^{*}\left({\mathcal{H}}_2\right)$. Let ${\mathcal{H}}_{1,i},{\mathcal{H}}_{2,i}$ be Hilbert $\mathcal{A}$-modules and let ${\Lambda }_i\in {\mathrm{End}}_{\mathcal{A}}^{*}(W_i,{\mathcal{H}}_{1,i})$, ${\Gamma }_i\in {\mathrm{End}}_{\mathcal{A}}^{*}(V_i,{\mathcal{H}}_{2,i})$ be adjointable maps. Assume that ${(W_i,{\Lambda }_i,v_i)}_{i\in I}$ is a $K_1$-g-fusion frame for ${\mathcal{H}}_1$ with bounds $A_1,B_1>0$ and that ${(V_i,{\Gamma }_i,v_i)}_{i\in I}$ is a $K_2$-g-fusion frame for ${\mathcal{H}}_2$ with bounds $A_2,B_2>0$, where $K_i\in {\mathrm{End}}_{\mathcal{A}}^{*}\left({\mathcal{H}}_i\right)$ ($i=1,2$). Define, for each $i\in I$,

$${\Theta }_i:{\mathcal{H}}_1\oplus {\mathcal{H}}_2⟶{\mathcal{H}}_{1,i}\oplus {\mathcal{H}}_{2,i},{\Theta }_i(x,y)=\left({\Lambda }_i{P}_{W_i}x,{\Gamma }_i{P}_{V_i}y\right).$$

Then we have the following theorem:

Theorem 1.  

Assume that ${(W_i,{\Lambda }_i,v_i)}_{i\in I}$ is a $K_1$-g-fusion frame for ${\mathcal{H}}_1$ with bounds $A_1,B_1>0$ and that ${(V_i,{\Gamma }_i,v_i)}_{i\in I}$ is a $K_2$-g-fusion frame for ${\mathcal{H}}_2$ with bounds $A_2,B_2>0$, then the family ${\left\{(W_i\oplus V_i,{\Theta }_i,v_i)\right\}}_{i\in I}$ is a $(K_1\oplus K_2)$-g-fusion frame for ${\mathcal{H}}_1\oplus {\mathcal{H}}_2$ with bounds $A=min\{A_1,A_2\}$ and $B=max\{B_1,B_2\}$. Furthermore, if $T^{\left(1\right)}$ and $T^{\left(2\right)}$ are respectively the synthesis operators of ${(W_i,{\Lambda }_i,v_i)}_{i\in I}$ and ${(V_i,{\Gamma }_i,v_i)}_{i\in I}$ and $S^{\left(1\right)}$ and $S^{\left(2\right)}$ their frame operators, then the synthesis operator T of the product satisfies $T=T^{\left(1\right)}\oplus T^{\left(2\right)}$, and the frame operator satisfies $S=S^{\left(1\right)}\oplus S^{\left(2\right)}$.

Proof. 

Since ${(W_i,{\Lambda }_i,v_i)}_{i\in ì}$ is a $K_1$-g-fusion frame for ${\mathcal{H}}_1$, it is in particular a Bessel family. Thus, there exists a scalar $B_1>0$ such that for all $x\in {\mathcal{H}}_1$

$$\sum _{i\in I}{v}_i^2\langle {\Lambda }_i{P}_{W_i}x,{\Lambda }_i{P}_{W_i}x\rangle \le B_1\langle x,x\rangle ,$$

where the series converges in norm in $\mathcal{A}$. Similarly, for $(V_i,{\Gamma }_i,v_i)$ there is $B_2>0$ with analogous norm-convergent series

$$\sum _{i\in I}{v}_i^2\langle {\Gamma }_i{P}_{V_i}y,{\Gamma }_i{P}_{V_i}y\rangle \le B_2\langle y,y\rangle ,$$

for all $y\in {\mathcal{H}}_2$. Therefore for any $(x,y)\in {\mathcal{H}}_1\oplus {\mathcal{H}}_2$,

$$\begin{array}{cc}\hfill \sum _{i\in I}{v}_i^2\langle {\Theta }_i(x,y),{\Theta }_i(x,y)\rangle & =\sum _{i\in I}{v}_i^2\left(\langle {\Lambda }_i{P}_{W_i}x,{\Lambda }_i{P}_{W_i}x\rangle +\langle {\Gamma }_i{P}_{V_i}y,{\Gamma }_i{P}_{V_i}y\rangle \right)\hfill \\ \hfill & =\left(\sum _{i\in I}{v}_i^2\langle {\Lambda }_i{P}_{W_i}x,{\Lambda }_i{P}_{W_i}x\rangle \right)+\left(\sum _{i\in I}{v}_i^2\langle {\Gamma }_i{P}_{V_i}y,{\Gamma }_i{P}_{V_i}y\rangle \right),\hfill \end{array}$$

each summand being norm convergent in $\mathcal{A}$; hence the whole sum converges in norm. Moreover

$$\sum _{i\in I}{v}_i^2\langle {\Theta }_i(x,y),{\Theta }_i(x,y)\rangle \le B_1\langle x,x\rangle +B_2\langle y,y\rangle \le max\{B_1,B_2\}\langle (x,y),(x,y)\rangle$$

shows the desired uniform Bessel bound on the product.

By the $K_1$-g-fusion inequality on ${\mathcal{H}}_1$ we have the $\mathcal{A}$-valued inequality

$$A_1\langle K_1^{*}x,K_1^{*}x\rangle \le \sum _{i\in I}{v}_i^2\langle {\Lambda }_i{P}_{W_i}x,{\Lambda }_i{P}_{W_i}x\rangle \le B_1\langle x,x\rangle ,$$

and similarly for ${\mathcal{H}}_2$:

$$A_2\langle K_2^{*}y,K_2^{*}y\rangle \le \sum _{i\in I}{v}_i^2\langle {\Gamma }_i{P}_{V_i}y,{\Gamma }_i{P}_{V_i}y\rangle \le B_2\langle y,y\rangle .$$

We can see that these two $\mathcal{A}$-valued inequalities yields

$$A_1\langle K_1^{*}x,K_1^{*}x\rangle +A_2\langle K_2^{*}y,K_2^{*}y\rangle \le \sum _{i\in I}{v}_i^2\langle {\Theta }_i(x,y),{\Theta }_i(x,y)\rangle \le B_1\langle x,x\rangle +B_2\langle y,y\rangle .$$

Since $A_1,A_2$ are positive scalars,

$$A_1\langle K_1^{*}x,K_1^{*}x\rangle +A_2\langle K_2^{*}y,K_2^{*}y\rangle \ge min\{A_1,A_2\}\left(\langle K_1^{*}x,K_1^{*}x\rangle +\langle K_2^{*}y,K_2^{*}y\rangle \right),$$

and likewise

$$B_1\langle x,x\rangle +B_2\langle y,y\rangle \le max\{B_1,B_2\}\left(\langle x,x\rangle +\langle y,y\rangle \right).$$

Observing that ${(K_1\oplus K_2)}^{*}(x,y)=(K_1^{*}x,K_2^{*}y)$ and that $\langle (x,y),(x,y)\rangle =\langle x,x\rangle +\langle y,y\rangle$, we obtain the claimed inequalities with $A=min\{A_1,A_2\}$ and $B=max\{B_1,B_2\}$, that is

$$A\langle {(K_1\oplus K_2)}^{*}(x,y),{(K_1\oplus K_2)}^{*}(x,y)\rangle \le \sum _{i\in I}{v}_i^2\langle {\Theta }_i(x,y),{\Theta }_i(x,y)\rangle \le B\langle (x,y),(x,y)\rangle .$$

Now, compute the adjoint ${\Theta }_i^{*}:{\mathcal{H}}_{1,i}\oplus {\mathcal{H}}_{2,i}\to {\mathcal{H}}_1\oplus {\mathcal{H}}_2$. For $(u,v)\in {\mathcal{H}}_{1,i}\oplus {\mathcal{H}}_{2,i}$ and $(x,y)\in {\mathcal{H}}_1\oplus {\mathcal{H}}_2$ one has

$$\langle {\Theta }_i(x,y),(u,v)\rangle ={\langle {\Lambda }_i{P}_{W_i}x,u\rangle }_{{\mathcal{H}}_{1,i}}+{\langle {\Gamma }_i{P}_{V_i}y,v\rangle }_{{\mathcal{H}}_{2,i}}=\langle (x,y),(P_{W_i}{\Lambda }_i^{*}u,P_{V_i}{\Gamma }_i^{*}v)\rangle ,$$

hence

$${\Theta }_i^{*}(u,v)=\left(P_{W_i}{\Lambda }_i^{*}u,P_{V_i}{\Gamma }_i^{*}v\right).$$

Therefore the operator ${\Theta }_i^{*}{\Theta }_i$ acts on $(x,y)$ by

$${\Theta }_i^{*}{\Theta }_i(x,y)=\left(P_{W_i}{\Lambda }_i^{*}{\Lambda }_i{P}_{W_i}x,P_{V_i}{\Gamma }_i^{*}{\Gamma }_i{P}_{V_i}y\right).$$

Multiplying by the scalar weight $v_i^2$ and summing over i gives the frame operator on the product:

$$S(x,y)=\sum _{i\in I}{v}_i^2{\Theta }_i^{*}{\Theta }_i(x,y)=\left(\sum _{i\in I}{v}_i^2{P}_{W_i}{\Lambda }_i^{*}{\Lambda }_i{P}_{W_i}x,\sum _{i\in I}{v}_i^2{P}_{V_i}{\Gamma }_i^{*}{\Gamma }_i{P}_{V_i}y\right).$$

The right-hand side is precisely $(S^{\left(1\right)}x,S^{\left(2\right)}y)$ where

$$S^{\left(1\right)}=\sum _i{v}_i^2{P}_{W_i}{\Lambda }_i^{*}{\Lambda }_i{P}_{W_i}and{S}^{\left(2\right)}=\sum _i{v}_i^2{P}_{V_i}{\Gamma }_i^{*}{\Gamma }_i{P}_{V_i}$$

are the frame operators of the component families. Thus $S=S^{\left(1\right)}\oplus S^{\left(2\right)}$. In particular S is positive and the operator inequalities $A\left(K_1{K}_1^{*}\oplus K_2{K}_2^{*}\right)\le S\le B{I}_{{\mathcal{H}}_1\oplus {\mathcal{H}}_2}$ hold in ${\mathrm{End}}_{\mathcal{A}}^{*}({\mathcal{H}}_1\oplus {\mathcal{H}}_2)$. This completes the proof. □

Example 1.  

Let $\mathcal{A}={\mathbb{C}}^2$ with coordinate-wise operations and the usual involution. Consider the left $\mathcal{A}$-modules ${\mathcal{H}}_1={\mathcal{H}}_2={\mathcal{A}}^6.$ Any element of ${\mathcal{H}}_1$ or ${\mathcal{H}}_2$, can be written as $x=(x_1,\dots ,x_6)$, where $x_m=(a_m,b_m)\in \mathcal{A}={\mathbb{C}}^2$, for $m=1,...,6$. The $\mathcal{A}$-valued inner product is given by

$$\langle x,y\rangle =\left(\sum _{m=1}^6{a}_m\overline{c_m},\sum _{m=1}^6{b}_m\overline{d_m}\right),$$

for $y=(y_1,\dots ,y_6)$ with $y_m=(c_m,d_m)$. In particular,

$$\langle x,x\rangle =\left(\sum _{m=1}^6|a_m{|}^2,\sum _{m=1}^6{\left|b_m\right|}^2\right).$$

Define two diagonal adjointable operators $K_1$ and $K_2$ on ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ respectively by:

$$K_1(x_1,\dots ,x_6)=(x_1,2x_2,x_3,2x_4,x_5,2x_6),$$

$$K_2(y_1,\dots ,y_6)=(y_1,3y_2,y_3,3y_4,y_5,3y_6).$$

Both are self-adjoint, so $K_i^{*}=K_i\in {\mathrm{End}}_{\mathcal{A}}\left({\mathcal{H}}_i\right)$ for $i=1,2$.

Denote $e_1=(1,0,0,0,0,0)$, ... , $e_6=(0,0,0,0,0,1)$ the canonical elements of ${\mathcal{H}}_j$, $j=1,2$. For the Hilbert ${\mathrm{C}}^{*}$-module ${\mathcal{H}}_1$ define

$$W_1={\mathrm{span}}_{\mathcal{A}}\{e_1,e_2\},W_2={\mathrm{span}}_{\mathcal{A}}\{e_3,e_4\},W_3={\mathrm{span}}_{\mathcal{A}}\{e_5,e_6\},$$

and let

$${\Lambda }_i:W_i\to {\mathcal{A}}^2,{\Lambda }_i(x_{2i-1},x_{2i})=(x_{2i-1},x_{2i}).$$

For ${\mathcal{H}}_2$ choose the submodules:

$$V_1={\mathrm{span}}_{\mathcal{A}}\{e_1,e_3\},V_2={\mathrm{span}}_{\mathcal{A}}\{e_2,e_5\},V_3={\mathrm{span}}_{\mathcal{A}}\{e_4,e_6\},$$

and define

$${\Gamma }_i:V_i\to {\mathcal{A}}^2,{\Gamma }_i(x_j,x_k)=(x_j,x_k).$$

All weights $v_i$, $i=1,2,3$ are choosing equal to 1.

Now observe that for all $x\in {\mathcal{H}}_1$ and all $y\in {\mathcal{H}}_2$ we have

$$\begin{array}{ccccc}{\displaystyle \frac{1}{4}}\langle K_1^{*}x,K_1^{*}x\rangle \hfill & \le \hfill & \sum _{i=1}^3\langle {\Lambda }_i{P}_{W_i}x,{\Lambda }_i{P}_{W_i}x\rangle \hfill & =\hfill & \langle x,x\rangle ,\hfill \\ {\displaystyle \frac{1}{9}}\langle K_2^{*}y,K_2^{*}y\rangle \hfill & \le \hfill & \sum _{i=1}^3\langle {\Gamma }_i{P}_{V_i}y,{\Gamma }_i{P}_{V_i}y\rangle \hfill & =\hfill & \langle y,y\rangle .\hfill \end{array}$$

So $(V_i,{\Gamma }_i)$ is a $K_2$-g-fusion frame with bounds $A_2=\frac{1}{9},B_2=1$.

The product family

$${\left\{(W_i\oplus V_i,{\Theta }_i,1)\right\}}_{i=1}^3,{\Theta }_i(x,y)=({\Lambda }_i{P}_{W_i}x,{\Gamma }_i{P}_{V_i}y),$$

is a $(K_1\oplus K_2)$-g-fusion frame for ${\mathcal{H}}_1\oplus {\mathcal{H}}_2$ with bounds

$$A=min\{A_1,A_2\}=\frac{1}{9},B=max\{B_1,B_2\}=1.$$

That is, for all $(x,y)\in {\mathcal{H}}_1\oplus {\mathcal{H}}_2$,

$${\displaystyle \frac{1}{9}}\langle {(K_1\oplus K_2)}^{*}(x,y),{(K_1\oplus K_2)}^{*}(x,y)\rangle \le \sum _{i=1}^3\langle {\Theta }_i(x,y),{\Theta }_i(x,y)\rangle \le \langle (x,y),(x,y)\rangle .$$

4. Perturbation Theorem

Let $K_i\in {\mathrm{End}}_{\mathcal{A}}^{*}\left({\mathcal{H}}_i\right)$ ($i=1,2$), and assume that ${\left\{(W_i,{\Lambda }_i,v_i)\right\}}_{i\in I}$ is a $K_1$-g-fusion frame for ${\mathcal{H}}_1$ with frame bounds $0<A_1\le B_1<\infty$, and that ${\left\{(V_i,{\Gamma }_i,v_i)\right\}}_{i\in I}$ is a $K_2$-g-fusion frame for ${\mathcal{H}}_2$ with frame bounds $0<A_2\le B_2<\infty$. Denote their product frame by $\mathcal{F}={\left\{(W_i\oplus V_i,{\Theta }_i,v_i)\right\}}_{i\in I}$ on $\mathcal{H}={\mathcal{H}}_1\oplus {\mathcal{H}}_2,$ by taking the common weights $v_i$ ($i\in I$). The following theorem gives a perturbation result saying that if each component family is a K-g-fusion frame and each component perturbation is small, then the perturbed product family is again a K-g-fusion frame on the direct sum ${\mathcal{H}}_1\oplus {\mathcal{H}}_2$.

Theorem 2.  

Let ${\mathcal{F}}^{\prime }={\left\{(W_i^{\prime },V_i^{\prime },{\Lambda }_i^{\prime },{\Gamma }_i^{\prime },v_i)\right\}}_{i\in I}$ be a perturbed family with the same weights $v_i$, where $W_i^{\prime }\subset {\mathcal{H}}_1$ and $V_i^{\prime }\subset {\mathcal{H}}_2$ are orthogonally complemented submodules, and ${\Lambda }_i^{\prime },{\Gamma }_i^{\prime }$ are adjointable operators. Assume there exist scalars $r_1,r_2>0$ such that, for all $x_1\in {\mathcal{H}}_1$ and $x_2\in {\mathcal{H}}_2$,

$$\left\{\begin{array}{cc}{\displaystyle \sum _{i\in I}}{v}_i^2\langle ({\Lambda }_i{P}_{W_i}-{\Lambda }_i^{\prime }{P}_{W_i^{\prime }})x_1,({\Lambda }_i{P}_{W_i}-{\Lambda }_i^{\prime }{P}_{W_i^{\prime }})x_1\rangle \hfill & \le r_1\langle K_1^{*}{x}_1,K_1^{*}{x}_1\rangle ,\hfill \\ \\ {\displaystyle \sum _{i\in I}}{v}_i^2\langle ({\Gamma }_i{P}_{V_i}-{\Gamma }_i^{\prime }{P}_{V_i^{\prime }})x_2,({\Gamma }_i{P}_{V_i}-{\Gamma }_i^{\prime }{P}_{V_i^{\prime }})x_2\rangle \hfill & \le r_2\langle K_2^{*}{x}_2,K_2^{*}{x}_2\rangle .\hfill \end{array}\right.$$

(1)

If $A_1>r_1$ and $A_2>r_2$, then the perturbed product frame ${\mathcal{F}}^{\prime }$ is a $(K_1\oplus K_2)$-g-fusion frame for $\mathcal{H}$ with frame bounds $A^{\prime }:=min\{{(\sqrt{A_1}-\sqrt{r_1})}^2,{(\sqrt{A_2}-\sqrt{r_2})}^2\}$ and $B^{\prime }:=max\{2B_1+2r_1\parallel K_1^{*}{\parallel }^2,2B_2+2r_2\parallel K_2^{*}{\parallel }^2\}$.

Proof. 

For $x_1\in {\mathcal{H}}_1$ note that ${\Lambda }_i^{\prime }{P}_{W_i^{\prime }}{x}_1={\Lambda }_i{P}_{W_i}{x}_1+\left({\Lambda }_i^{\prime }{P}_{W_i^{\prime }}-{\Lambda }_i{P}_{W_i}\right)x_1$. For $x_2\in {\mathcal{H}}_2$ set similarly ${\Gamma }_i^{\prime }{P}_{V_i^{\prime }}{x}_2={\Gamma }_i{P}_{V_i}{x}_2+\left({\Gamma }_i^{\prime }{P}_{V_i^{\prime }}-{\Gamma }_i{P}_{V_i}\right)x_2$.

First, from the two g-fusion frame inequalities (1) and Theorem 1 we obtain immediately, with $A=min\{A_1,A_2\}$ and $B=max\{B_1,B_2\}$, that the product family $\mathcal{F}$ is a $(K_1\oplus K_2)$-g-fusion frame with bounds $A,B$. Thus satisfies

$$\begin{array}{ccc}A\langle {(K_1\oplus K_2)}^{*}(x_1,x_2),{(K_1\oplus K_2)}^{*}(x_1,x_2)\rangle \hfill & \le \hfill & {\displaystyle \sum _{i\in I}}{v}_i^2\langle {\Theta }_i(x_1,x_2),{\Theta }_i(x_1,x_2)\rangle \hfill \\ & =\hfill & {\displaystyle \sum _{i\in I}}{v}_i^2\left(\langle {\Lambda }_i{P}_{W_i}{x}_1,{\Lambda }_i{P}_{W_i}{x}_1\rangle +\langle {\Gamma }_i{P}_{V_i}{x}_2,{\Gamma }_i{P}_{V_i}{x}_2\rangle \right)\hfill \\ & \le \hfill & B\langle (x_1,x_2),(x_1,x_2)\rangle .\hfill \end{array}$$

Now, fix $x_1\in {\mathcal{H}}_1$. Consider the perturbed left-component sum

$$\begin{array}{ccc}{S}_1^{\prime }\left(x_1\right)\hfill & :=\hfill & {\displaystyle \sum _{i\in I}}{v}_i^2\langle {\Lambda }_i^{\prime }{P}_{W_i^{\prime }}{x}_1,{\Lambda }_i^{\prime }{P}_{W_i^{\prime }}{x}_1\rangle \hfill \\ & =\hfill & {\displaystyle \sum _{i\in I}}{v}_i^2\langle {\Lambda }_i{P}_{W_i}{x}_1+\left({\Lambda }_i^{\prime }{P}_{W_i^{\prime }}-{\Lambda }_i{P}_{W_i}\right)x_1,{\Lambda }_i{P}_{W_i}{x}_1+\left({\Lambda }_i^{\prime }{P}_{W_i^{\prime }}-{\Lambda }_i{P}_{W_i}\right)x_1\rangle .\hfill \end{array}$$

For the sake of readability, we denote X and Y the elements of the Hilbert ${\mathrm{C}}^{*}$-module ${\ell }^2(I,\mathcal{A})$ defined by

$$X:={\left(v_i{\Lambda }_i{P}_{W_i}{x}_1\right)}_{i\in I},Y:={\left(v_i\left({\Lambda }_i^{\prime }{P}_{W_i^{\prime }}-{\Lambda }_i{P}_{W_i}\right)x_1\right)}_{i\in I}.$$

Thus

$$S_1^{\prime }\left(x_1\right)={\langle X,X\rangle }_{{\ell }^2(I,\mathcal{A})}+{\langle Y,Y\rangle }_{{\ell }^2(I,\mathcal{A})}+{\langle X,Y\rangle }_{{\ell }^2(I,\mathcal{A})}+{\langle X,Y\rangle }_{{\ell }^2(I,\mathcal{A})}^{*}.$$

(2)

This follows from the computations:

$${\langle X,Y\rangle }_{{\ell }^2(I,\mathcal{A})}+{\langle X,Y\rangle }_{{\ell }^2(I,\mathcal{A})}^{*}=\sum _{i\in I}{v}_i^2\left(\langle {\Lambda }_i{P}_{W_i}{x}_1,\left({\Lambda }_i^{\prime }{P}_{W_i^{\prime }}-{\Lambda }_i{P}_{W_i}\right)x_1\rangle +\langle \left({\Lambda }_i^{\prime }{P}_{W_i^{\prime }}-{\Lambda }_i{P}_{W_i}\right)x_1,{\Lambda }_i{P}_{W_i}{x}_1\rangle \right)$$

$${\langle X,X\rangle }_{{\ell }^2(I,\mathcal{A})}=\sum _{i\in I}{v}_i^2\langle {\Lambda }_i{P}_{W_i}{x}_1,{\Lambda }_i{P}_{W_i}{x}_1\rangle$$

$${\langle Y,Y\rangle }_{{\ell }^2(I,\mathcal{A})}=\sum _{i\in I}{v}_i^2\langle \left({\Lambda }_i^{\prime }{P}_{W_i^{\prime }}-{\Lambda }_i{P}_{W_i}\right)x_1,\left({\Lambda }_i^{\prime }{P}_{W_i^{\prime }}-{\Lambda }_i{P}_{W_i}\right)x_1\rangle ,$$

Now, since ${\langle X+Y,X+Y\rangle }_{{\ell }^2(I,\mathcal{A})}$ and ${\langle X-Y,X-Y\rangle }_{{\ell }^2(I,\mathcal{A})}$ are non negative in $\mathcal{A}$, we have

$$\begin{array}{ccc}-\sqrt{{\displaystyle \frac{r_1}{A_1}}}{\langle X,X\rangle }_{{\ell }^2(I,\mathcal{A})}-\sqrt{{\displaystyle \frac{A_1}{r_1}}}{\langle Y,Y\rangle }_{{\ell }^2(I,\mathcal{A})}\hfill & \le \hfill & {\langle X,Y\rangle }_{{\ell }^2(I,\mathcal{A})}+{\langle X,Y\rangle }_{{\ell }^2(I,\mathcal{A})}^{*}\hfill \\ \hfill & \le \hfill & {\langle X,X\rangle }_{{\ell }^2(I,\mathcal{A})}+{\langle Y,Y\rangle }_{{\ell }^2(I,\mathcal{A})}\hfill \end{array}$$

(3)

Since $S_1^{\prime }\left(x_1\right)={\langle X+Y,X+Y\rangle }_{{\ell }^2(I,\mathcal{A})}$ we deduce from (3) and (2) that

$$\begin{array}{ccc}{S}_1^{\prime }\left(x_1\right)\hfill & \le \hfill & 2{\langle X,X\rangle }_{{\ell }^2(I,\mathcal{A})}+2{\langle Y,Y\rangle }_{{\ell }^2(I,\mathcal{A})}\hfill \\ & \le \hfill & 2B_1\langle x_1,x_1\rangle +2r_1{\parallel K_1^{*}\parallel }^2\langle x_1,x_1\rangle \hfill \\ & =\hfill & (2B_1+2r_1\parallel K_1^{*}{\parallel }^2)\langle x_1,x_1\rangle \hfill \end{array}$$

For the other inequality, using (1), (2), (3) and the hypothesis $\sqrt{{\displaystyle \frac{r_1}{A_1}}}<1$, we obtain that

$$\begin{array}{ccc}{S}_1^{\prime }\left(x_1\right)\hfill & \ge \hfill & {\langle X,X\rangle }_{{\ell }^2(I,\mathcal{A})}+{\langle Y,Y\rangle }_{{\ell }^2(I,\mathcal{A})}-\sqrt{{\displaystyle \frac{r_1}{A_1}}}{\langle X,X\rangle }_{{\ell }^2(I,\mathcal{A})}-\sqrt{{\displaystyle \frac{A_1}{r_1}}}{\langle Y,Y\rangle }_{{\ell }^2(I,\mathcal{A})}\hfill \\ & =\hfill & (1-\sqrt{{\displaystyle \frac{r_1}{A_1}}}){\langle X,X\rangle }_{{\ell }^2(I,\mathcal{A})}-(\sqrt{{\displaystyle \frac{A_1}{r_1}}}-1)){\langle Y,Y\rangle }_{{\ell }^2(I,\mathcal{A})}\hfill \\ & \ge \hfill & (A_1(1-\sqrt{{\displaystyle \frac{r_1}{A_1}}})\langle K_1^{*}{x}_1,K_1^{*}{x}_1\rangle -r_1(\sqrt{{\displaystyle \frac{A_1}{r_1}}}-1))\langle K_1^{*}{x}_1,K_1^{*}{x}_1\rangle \hfill \\ & \ge \hfill & (A_1(1-\sqrt{{\displaystyle \frac{r_1}{A_1}}})+r_1(1-\sqrt{{\displaystyle \frac{A_1}{r_1}}}))\langle K_1^{*}{x}_1,K_1^{*}{x}_1\rangle \hfill \\ & =\hfill & {(\sqrt{A_1}-\sqrt{r_1})}^2\langle K_1^{*}{x}_1,K_1^{*}{x}_1\rangle \hfill \end{array}$$

The same argument applied to the second component yields

$${(\sqrt{A_2}-\sqrt{r_2})}^2\langle K_2^{*}{x}_2,K_2^{*}{x}_2\rangle \le S_2^{\prime }\left(x_2\right)\le (2B_2+2r_2\parallel K_2^{*}{\parallel }^2)\langle x_2,x_2\rangle .$$

Finally, For $(x_1,x_2)\in {\mathcal{H}}_1\oplus {\mathcal{H}}_2$ we have

$$S^{\prime }(x_1,x_2):=\sum _i{v}_i^2\langle {\Theta }_i^{\prime }(x_1,x_2),{\Theta }_i^{\prime }(x_1,x_2)\rangle =S_1^{\prime }\left(x_1\right)+S_2^{\prime }\left(x_2\right).$$

Thus combining the componentwise upper bounds yields

$$\begin{array}{ccc}{S}^{\prime }(x_1,x_2)\hfill & \le \hfill & (2B_1+2r_1\parallel K_1^{*}{\parallel }^2)\langle x_1,x_1\rangle +(2B_2+2r_2\parallel K_2^{*}{\parallel }^2)\langle x_2,x_2\rangle \hfill \\ & \le \hfill & B^{\prime }\langle (x_1,x_2),(x_1,x_2)\rangle ,\hfill \end{array}$$

with $B^{\prime }=max\left\{\right(2B_1+2r_1\parallel K_1^{*}{\parallel }^2),(2B_2+2r_2\parallel K_2^{*}{\parallel }^2\left)\right\}$.

Similarly combining the lower bounds we get

$$\begin{array}{ccc}{S}^{\prime }(x_1,x_2)\hfill & \ge \hfill & {(\sqrt{A_1}-\sqrt{r_1})}^2\langle K_1^{*}{x}_1,K_1^{*}{x}_1\rangle +{(\sqrt{A_2}-\sqrt{r_2})}^2\langle K_2^{*}{x}_2,K_2^{*}{x}_2\rangle \hfill \\ & \ge \hfill & A^{\prime }\langle {(K_1\oplus K_2)}^{*}(x_1,x_2),{(K_1\oplus K_2)}^{*}(x_1,x_2)\rangle ,\hfill \end{array}$$

with $A^{\prime }=min\{{(\sqrt{A_1}-\sqrt{r_1})}^2,{(\sqrt{A_2}-\sqrt{r_2})}^2\}$. This yields the claimed two-sided $\mathcal{A}$-order inequalities and completes the proof that ${\mathcal{F}}^{\prime }$ is a $(K_1\oplus K_2)$-g-fusion frame with bounds $A^{\prime },B^{\prime }$. □