Controlled $g$-frames in Hilbert $C^*$-modules

TL;DR

This paper extends the concept of controlled frames to g-frames within Hilbert C*-modules, providing operator theoretic characterizations, relationships, and perturbation results to enhance understanding and potential applications.

Contribution

It introduces controlled g-frames in Hilbert C*-modules, establishing their characterizations, relationships with g-frames, and perturbation properties, advancing the theoretical framework.

Findings

Equivalent conditions for controlled g-frames established

Operator theoretic characterizations provided

Perturbation results for controlled g-frames proved

Abstract

To improve the numerical efficiency of iterative algorithms for inverting the frame operator, the controlled frame was introduced by Balazs et al. \cite{Balazs}, and has since been given more importance. In this paper, we introduce the concept of controlled g-frames in Hilbert $C^{*}$-modules. We establish the equivalent condition for controlled $g$-frame using operator theoretic approach. We investigate some operator theoretic characterizations of controlled $g$-frames and controlled $g$-Bessel sequences. We also established the relationship between $g$-frames and controlled $g$-frames in Hilbert $C^{*}$-modules. At the end we prove some perturbation results on controlled $g$-frames.

Equations

$$C⟨f,f⟩\le \sum _{j\in \mathbb{J}}⟨f,f_j⟩⟨f_j,f⟩\le D⟨f,f⟩,\forall f\in \mathcal{H}.$$

$$C⟨f,f⟩\le \sum _{j\in \mathbb{J}}⟨{\wedge }_{j}f,{\wedge }_{j}f⟩\le D⟨f,f⟩,\forall f\in \mathcal{H}.$$

$$S_{\Lambda }f=\sum _{j\in J}{\Lambda }_j^{\ast }{\Lambda }_{j}f.$$

$$m⟨f,f⟩\le \sum _{j\in \mathbb{J}}⟨f,f_j⟩⟨C{f}_j,f⟩\le M⟨f,f⟩,\forall f\in \mathcal{H}.$$

$$A⟨f,f⟩\le \sum _{j\in J}⟨{\Lambda }_{j}Cf,{\Lambda }_j{C}^{\prime }f⟩\le B⟨f,f⟩,\forall f\in \mathcal{H}.$$

$$\sum _{j\in J}⟨{\Lambda }_{j}Cf,{\Lambda }_j{C}^{\prime }f⟩\le B⟨f,f⟩,\forall f\in \mathcal{H}.$$

$$\displaystyle\Lambda_{j}f=\sum_{k=1}^{j}\big{\langle}f,\frac{e_{j}}{\sqrt{j}}\big{\rangle}e_{k}.$$

$$\displaystyle\Lambda_{j}^{*}(g)=\sum_{k=1}^{j}\big{\langle}f,\frac{e_{k}}{\sqrt{j}}\big{\rangle}e_{j}.$$

$$\displaystyle\Big{\langle}\Lambda_{j}Cf,\Lambda_{j}C^{\prime}f\Big{\rangle}$$

$$\displaystyle\sum_{j=1}^{\infty}\Big{\langle}\Lambda_{j}Cf,\Lambda_{j}C^{\prime}f\Big{\rangle}$$

$$T_{(C,C^{\prime })}(\{g_j{\}}_{j\in J})=\sum _{j\in J}(C{C}^{\prime }{)}^{\frac{1}{2}}{\Lambda }_j^{\ast }{g}_j,\forall \{g_j{\}}_{j\in J}\in l^2(\{{\mathcal{H}}_j{\}}_{j\in J})$$

$$\displaystyle T^{*}_{(C,C^{\prime})}(f)=\big{\{}\Lambda_{j}(C^{\prime}C)^{\frac{1}{2}}f\big{\}}_{j\in J}$$

$$S_{(C,C^{\prime })}f$$

$$\displaystyle\sum_{j\in J}\langle\Lambda_{j}Cf,\Lambda_{j}C^{\prime}f\rangle=\big{\langle}\sum_{j\in J}C^{\prime}\Lambda^{*}_{j}\Lambda_{j}Cf,f\big{\rangle}=\big{\langle}S_{(C,C^{\prime})}f,f\big{\rangle},$$

$$\displaystyle\big{\langle}S_{(C,C^{\prime})}f,g\big{\rangle}$$

$$\displaystyle A\langle f,f\rangle\leq\big{\langle}S_{(C,C^{\prime})}f,f\big{\rangle}\leq B\langle f,f\rangle$$

$$A\Vert f{\Vert }^2\le \Vert \sum _{j\in J}⟨{\Lambda }_{j}Cf,{\Lambda }_j{C}^{\prime }f⟩\Vert \le B\Vert f{\Vert }^2,\forall f\in \mathcal{H}.$$

$$A⟨f,f⟩\le \sum _{j\in J}⟨{\Lambda }_{j}Cf,{\Lambda }_j{C}^{\prime }f⟩\le B⟨f,f⟩,\forall f\in \mathcal{H}.$$

$$\Vert A⟨f,f⟩\Vert \le \Vert \sum _{j\in J}⟨{\Lambda }_{j}Cf,{\Lambda }_j{C}^{\prime }f⟩\Vert \le \Vert B⟨f,f⟩\Vert ,\forall f\in \mathcal{H}$$

$$A\Vert f{\Vert }^2\le \Vert \sum _{j\in J}⟨{\Lambda }_{j}Cf,{\Lambda }_j{C}^{\prime }f⟩\Vert \le B\Vert f{\Vert }^2,\forall f\in \mathcal{H}.$$

$$\displaystyle\big{\langle}S_{(c,c^{\prime})}^{\frac{1}{2}}f,S_{(c,c^{\prime})}^{\frac{1}{2}}f\big{\rangle}=\big{\langle}S_{(c,c^{\prime})}f,f\big{\rangle}=\sum_{j\in J}\langle\Lambda_{j}Cf,\Lambda_{j}C^{\prime}f\rangle.$$

$$\sqrt{A}\Vert f\Vert \le \Vert S_{(c,c^{\prime })}^{\frac{1}{2}}f\Vert \le \sqrt{B}\Vert f\Vert ,\forall f\in \mathcal{H}.$$

$$\displaystyle m\langle f,f\rangle\leq\big{\langle}S_{(c,c^{\prime})}^{\frac{1}{2}}f,S_{(c,c^{\prime})}^{\frac{1}{2}}f\big{\rangle}=\sum_{j\in J}\langle\Lambda_{j}Cf,\Lambda_{j}C^{\prime}f\rangle\leq M\langle f,f\rangle,~{}~{}\forall f\in\mathcal{H}.$$

$$A⟨f,f⟩\le \sum _{j\in J}⟨{\Lambda }_{j}Cf,{\Lambda }_{j}Cf⟩\le B⟨f,f⟩,\forall f\in \mathcal{H},$$

$$A\Vert f{\Vert }^2\le \Vert \sum _{j\in J}⟨{\Lambda }_{j}Cf,{\Lambda }_{j}Cf⟩\Vert \le B\Vert f{\Vert }^2,\forall f\in \mathcal{H}.$$

$$\displaystyle A\|f\|^{2}\leq\big{\|}\sum_{j\in J}\langle\Lambda_{j}f,\Lambda_{j}f\rangle\big{\|}\leq B\|f\|^{2},~{}~{}\forall f\in\mathcal{H}.$$

$$A\Vert f{\Vert }^2\le \Vert \sum _{j\in J}⟨{\Lambda }_{j}Cf,{\Lambda }_{j}Cf⟩\Vert \le B\Vert f{\Vert }^2,\forall f\in \mathcal{H}.$$

$$A\Vert f{\Vert }^2=A\Vert C{C}^{-1}f{\Vert }^2$$

$$A\Vert C{\Vert }^{-2}\Vert f{\Vert }^2\le \Vert \sum _{j\in J}⟨{\Lambda }_{j}f,{\Lambda }_{j}f⟩\Vert .$$

$$\Vert \sum _{j\in J}⟨{\Lambda }_{j}f,{\Lambda }_{j}f⟩\Vert$$

Full text

**Keywords: Hilbert $C^{*}$-module, Frame, $g$-frame, Controlled $g$-frame

**

MSC 2010: 42C15, 06D22, 46H35.

1 Introduction

It was Duffin and Schaeffer [6] in 1952 who initiated the notion of frames while studying nonharmonic Fourier series. After a long gap, in 1986, Daubechies et al. [5] reintroduced the same notion and developed the theory of frames. In general, frame is nothing but a spanning set and what makes it interesting is the redundance of additional vectors than those in the basis. Due to its redundancy it becomes more applicable not only in theoretical point of view but also in various kinds of applications. Due to their rich structure the subject draws the attention of many mathematicians, physicists and engineers since it is largely applicable in signal processing [9], image processing [4], coding and communications [19], sampling [7, 8], numerical analysis, filter theory [3]. Now a days it is used in compressive sensing, data analysis and other areas.

Hilbert $C^{*}$-module is a wide category between Hilbert space and Banach space. It was Frank and Larson [10], who initiated the theory of frames in Hilbert $C^{*}$-modules. For more details of frames in Hilbert $C^{*}$-modules one may refer to the Doctoral Dissertation [15], Han et al. [12], Han et al. [13] and the references there in. The notion of g-frame or generalized frame in Hilbert $C^{*}$-module is introduced by Sun [20]. For more on $g$-frames one can refer to Khosravi and Khosravi [16], Fu and Zhang [11], Li and Leng [18]. To improve the numerical efficiency of iterative algorithms for inverting the frame operator, controlled frame was introduced by Balazs et al. [2] in Hilbert spaces. Recently, Kouchi and Rahimi [17] introduced Controlled frames in Hilbert $C^{*}$-modules. Motivated from the above literature, we introduce the notion of controlled g-frame in Hilbert $C^{*}$-modules.

2 Preliminaries

Let us briefly recall some definitions and basic properties of Hilbert $C^{*}$ modules. Hilbert $C^{*}$-modules are generalization of Hilbert spaces by allowing the inner product to take values in a $C^{*}$-algebra rather than the usual fields $\mathbb{R}$ or $\mathbb{C}$.

Let $\mathcal{A}$ be a unital $C^{*}$-algebra. The Hilbert $C^{*}$-module or Hilbert $\mathcal{A}$-module is defined as follows:

Definition 2.1**.**

Let $\mathcal{H}$ be a left $\mathcal{A}$-module such that the linear structure of $\mathcal{A}$ and $\mathcal{H}$ are compatible; $\mathcal{H}$ is called a pre-Hilbert $\mathcal{A}$-module if $\mathcal{H}$ is equipped with an $\mathcal{A}$-valued inner product $\langle\cdot,\cdot\rangle:\mathcal{H}\times\mathcal{H}\to\mathcal{A}$ such that

(i) $\langle f,f\rangle\geq 0$ $,\forall f\in\mathcal{H}$, and $\langle f,f\rangle=0$ if and only if $f=0$

(ii) $\langle f,g\rangle=\langle g,f\rangle^{*}$

(iii) $\langle af+g,h\rangle=a\langle f,h\rangle+\langle g,h\rangle$ for all $f,g,h\in\mathcal{H}$ and $a\in\mathcal{A}$.

For every $f\in\mathcal{H}$, the norm is defined as $\|f\|=\|\langle f,f\rangle\|^{\frac{1}{2}}$.

If $\mathcal{H}$ is complete with respect to the norm, it is called a Hilbert $\mathcal{A}$-module or a Hilbert $C^{*}$-module over $\mathcal{A}$.

Initially, it was Frank and Larson [10] who introduced the notion of frames in Hilbert $C^{*}$-modules. Their definition is the following:

Definition 2.2**.**

[10] A set of elements $\{f_{j}\}_{j\in\mathbb{J}}$ in a Hilbert $C^{*}$-module $\mathcal{H}$ over a unital $C^{*}$-algebra $\mathcal{A}$, is said to be a frame if there exist two constants $C,D>0$ such that

[TABLE]

After the introduction of frames in Hilbert $C^{*}$-modules, a lot of work on frame theory has been developed in Hilbert $C^{*}$-modules. The concept of g-frames in Hilbert $C^{*}$-modules was introduced by Sun [20]. The $g$-frame in Hilbert $C^{*}$-modules is defined as follows:

Definition 2.3**.**

[20] Let $\mathcal{H}$ be a Hilbert $C^{*}$-module over $\mathcal{A}$, $\{\mathcal{H}_{j}\}_{j\in\mathbb{J}}$ be a sequence of closed subspaces of $\mathcal{H}$. A sequence $\{\wedge_{j}\in End_{A}^{*}(\mathcal{H},\mathcal{H}_{j})\}$ is called a g-frame in $\mathcal{H}$ with respect to $\{\mathcal{H}_{j}\}_{j\in\mathbb{J}}$ if there exist positive constants $C,D$ such that

[TABLE]

The g-frame operator $S_{\Lambda}:H\rightarrow H$ is defined as

[TABLE]

To improve the numerical efficiency of iterative algorithms to find the inverse of frame operator, a new notion of frame was introduced, that is called as controlled frames. Controlled frames in Hilbert space is introduced by Balazs et al. [2] in 2010. Very recently, controlled frames in Hilbert $C^{*}$-modules is introduced by Kouchi and Rahimi [17]. The controlled frame in Hilbert $C^{*}$-modules is defined as follow:

Definition 2.4**.**

[17] Let $\mathcal{H}$ be a Hilbert $C^{*}$-module and $C\in GL(\mathcal{H})$. A family of vectors $\{f_{j}\in\mathcal{H}\}_{j\in\mathbb{J}}$ is said to be a controlled frame in $\mathcal{H}$ or $C$-Controlled frame in $\mathcal{H}$ if there exist constants $m,M>0$ such that

[TABLE]

In the next section we introduce the notion of Controlled g-frames in Hilbert $C^{*}$-modules. We study several characterizations of a controlled g-frame, equivalent formulation, its operator theoretic behavior, its relationship with the frames etc.. In the end we present some stability results on controlled $g$-frames.

3 Controlled g-frame

Let $\mathcal{H}$ be a $C^{*}$-module over a unital $C^{*}$-algebra $\mathcal{A}$ with $\mathcal{A}$-valued inner product $\langle.,.\rangle$ and norm $\|.\|$. Let $\{\mathcal{H}_{j}\}_{j\in J}$ be a sequence of closed submodules of $\mathcal{H}$, where $J$ is any index set. Also let $GL^{+}(\mathcal{H})$ be the set of all positive bounded linear invertible operators on $\mathcal{H}$ with bounded inverse.

Definition 3.1**.**

Let $C,C^{\prime}\in GL^{+}(\mathcal{H})$. A sequence $\{\Lambda_{j}\in End_{\mathcal{A}}^{*}(\mathcal{H},\mathcal{H}_{j}):j\in J\}$ is said to be a $(C,C^{\prime})$-controlled g-frame for $\mathcal{H}$ with respect to $\{\mathcal{H}_{j}\}_{j\in J}$ if there exist constants $0<A\leq B<\infty$ such that

[TABLE]

When $A=B$, the sequence $\{\Lambda_{j}\in End_{\mathcal{A}}^{*}(\mathcal{H},\mathcal{H}_{j}):j\in J\}$ is called $(C,C^{\prime})$-controlled tight g-frame, and when $A=B=1$, it is called a $(C,C^{\prime})$-controlled Parseval g-frame.

Definition 3.2**.**

A sequence $\{\Lambda_{j}\in End_{\mathcal{A}}^{*}(\mathcal{H},\mathcal{H}_{j}):j\in J\}$ is said to be a $(C,C^{\prime})$-controlled g-Bessel sequence for $\mathcal{H}$ with respect to $\{\mathcal{H}_{j}\}_{j\in J}$ if there exists constant $0<B<\infty$ such that

[TABLE]

Example 3.1**.**

Let $\mathcal{H}$ be an ordinary inner product space, $J=\mathbb{N}$, and $\{e_{j}\}_{j=1}^{\infty}$ be an orthonormal basis for Hilbert $\mathcal{C}$-module $\mathcal{H}$. We construct $\mathcal{H}_{j}$ as $\mathcal{H}_{j}=\overline{span}\{e_{1},e_{2},...,e_{j}\}$ for each $j\in\mathbb{N}$.

Define $\Lambda_{j}:\mathcal{H}\rightarrow\mathcal{H}_{j}$ by

[TABLE]

The adjoint operator $\Lambda_{j}^{*}:\mathcal{H}_{j}\rightarrow\mathcal{H}$ can be easily found as

[TABLE]

Let us define $Cf=2f$ and $C^{\prime}f=\frac{1}{2}f$. Then for any $f\in\mathcal{H}$, we can estimate

[TABLE]

Therefore, for any $f\in\mathcal{H}$,

[TABLE]

This shows that $\{\Lambda_{j}:j\in\mathbb{N}\}$ is a $(C,C^{\prime})$-controlled Parseval $g$-frame for $\mathcal{H}$ with respect to $\{H_{j}\}_{j\in\mathbb{N}}$.

Suppose that $\{\Lambda_{j}\in End_{\mathcal{A}}^{*}(\mathcal{H},\mathcal{H}_{j}):j\in J\}$ be a $(C,C^{\prime})$-controlled g-frame for the Hilbert $C^{*}$-module $\mathcal{H}$ with respect to $\{\mathcal{H}_{j}\}_{j\in J}$. The bounded linear operator $T_{(C,C^{\prime})}:l^{2}(\{\mathcal{H}_{j}\}_{j\in J})\rightarrow\mathcal{H}$ defined by

[TABLE]

is called the synthesis operator for the $(C,C^{\prime})$-controlled g-frame $\{\Lambda_{j}:j\in J\}$.

The adjoint operator $T^{*}_{(C,C^{\prime})}:\mathcal{H}\rightarrow l^{2}(\{\mathcal{H}_{j}\}_{j\in J})$ given by

[TABLE]

is called the analysis operator for the $(C,C^{\prime})$-controlled g-frame $\{\Lambda_{j}:j\in J\}$.

When $C$ and $C^{\prime}$ commute with each other, and commute with the operator $\Lambda_{j}^{*}\Lambda_{j}$ for each $j$, then the $(C,C^{\prime})$-controlled g-frame operator $S_{(C,C^{\prime})}:\mathcal{H}\rightarrow\mathcal{H}$ is defined as

[TABLE]

For the above result one is referred to Hua and Huang [14]. So from now on we assume that $C$ and $C^{\prime}$ commute with each other, and commute with the operator $\Lambda_{j}^{*}\Lambda_{j}$ for each $j$.

Proposition 3.1**.**

Let $\{\Lambda_{j}:j\in J\}$ be a $(C,C^{\prime})$-controlled g-frame for the Hilbert $C^{*}$-module $\mathcal{H}$ with respect to $\{\mathcal{H}_{j}\}_{j\in J}$. Then the $(C,C^{\prime})$-controlled g-frame operator $S_{(C,C^{\prime})}$ is positive, self adjoint and invertible.

Proof.

The frame operator $S_{(C,C^{\prime})}$ for the $(C,C^{\prime})$-controlled g-frame is $S_{(C,C^{\prime})}f=\sum_{j\in J}C^{\prime}\Lambda_{j}^{*}\Lambda_{j}Cf$. As $\{\Lambda_{j}:j\in J\}$ is a $(C,C^{\prime})$-controlled g-frame, and from the following identity,

[TABLE]

we clearly see that $S_{(C,C^{\prime})}$ is a positive operator. Also it is clearly bounded and linear. Again

[TABLE]

Hence $S_{(C,C^{\prime})}^{*}=S_{(C^{\prime},C)}$. Also as $C$ and $C^{\prime}$ commute with each other and commute with $\Lambda_{j}^{*}\Lambda_{j}$, we have $S_{(C,C^{\prime})}=S_{(C^{\prime},C)}$. So the controlled g-frame operator is self adjoint. Alternatively, this can also be directly obtained as $S_{(C,C^{\prime})}$ is a positive operator, and every positive operator is self adjoint.

From the controlled g-frame identity we have

[TABLE]

where $Id_{\mathcal{H}}$ is the identity operator in $\mathcal{H}$. Thus the controlled g-frame operator $S_{(C,C^{\prime})}$ is invertible. ∎

Lemma 3.1**.**

[1] Let $\mathcal{A}$ be a $C^{*}$-algebra. Let $U$ and $V$ be two Hilbert $\mathcal{A}$-modules and $T\in End_{\mathcal{A}}^{*}(U,V)$. Then the following statements are equivalent:

$T$ is surjective. 2. 2.

$T^{*}$ is bounded below with respect to norm i.e there exists $m>0$ such that $\|T^{*}f\|\geq m\|f\|$ for all $f\in U$. 3. 3.

$T^{*}$ is bounded below with respect to inner product i.e there exists $m>0$ such that $\langle T^{*}f,T^{*}f\rangle\geq m\langle f,f\rangle$ for all $f\in U$.

With the help of the above Lemma 3.1, we establish an equivalent definition of $(C,C^{\prime})$-controlled g-frame.

Theorem 3.1**.**

Let $\{\Lambda_{j}:j\in J\}\subset End^{*}_{A}(\mathcal{H},\mathcal{H}_{j})$ and $\sum_{j\in J}\langle\Lambda_{j}Cf,\Lambda_{j}C^{\prime}f\rangle$ converge in norm for any $f\in\mathcal{H}$. Then $\{\Lambda_{j}:j\in J\}$ is a $(C,C^{\prime})-$controlled g-frame for $\mathcal{H}$ with respect to $\{\mathcal{H}_{j}\}_{j\in J}$ if and only if there exists constants $A,B>0$ such that

[TABLE]

Proof.

Let $\{\Lambda_{j}:j\in J\}$ be a $(C,C^{\prime})-$controlled g-frame for $\mathcal{H}$ with respect to $\{\mathcal{H}_{j}\}_{j\in J}$ with bound $A$ and $B$. Hence we have

[TABLE]

Since $\langle f,f\rangle\geq 0,~{}~{}\forall f\in\mathcal{H}$, then we can take the norm on the left, middle and right terms of the above inequality (3.9). Thus we have

[TABLE]

Conversely, suppose that

[TABLE]

From Proposition (3.1), the $(C,C^{\prime})-$controlled g-frame operator $S_{(C,C^{\prime})}$ is positive, self adjoint and invertible. Hence

[TABLE]

Using (3.11) in (3.10), we get

[TABLE]

According to Lemma 3.1 and inequality (3.12), there exist constant $m,M>0$ such that

[TABLE]

Therefore, $\{\Lambda_{j}:j\in J\}$ is a $(C,C^{\prime})-$controlled g-frame for $\mathcal{H}$ with respect to $\{\mathcal{H}_{j}\}_{j\in J}$. ∎

Definition 3.3**.**

Let $C\in GL^{+}(\mathcal{H})$. The sequence $\{\Lambda_{j}\in End_{A}^{*}(\mathcal{H},\mathcal{H}_{j}):j\in J\}$ is said to be a $(C,C)-$controlled g-frame or $C^{2}-$controlled g-frame if there exist constants $0<A\leq B<\infty$ such that

[TABLE]

or equivalently,

[TABLE]

Using some tools from operator algebras, Xiao and Zeng [21] have proved the following equivalent characterization of g-frames in Hilbert $C^{*}$-modules.

Theorem 3.2**.**

[21] Let $\{\Lambda_{j}\in End_{A}^{*}(\mathcal{H},{\mathcal{H}}_{j}):j\in J\}$ and $\sum_{j\in J}\langle\Lambda_{j}f,\Lambda_{j}f\rangle$ converge in norm for $f\in\mathcal{H}$. Then $\{\Lambda_{j}:j\in J\}$ is a g-frame for $\mathcal{H}$ with respect to $\{\mathcal{H}_{j}\}_{j\in J}$ if and only if there exist constants $A,B>0$ such that

[TABLE]

The above result can be easily seen as a corollary of our Theorem 3.1, when we take $C=C^{\prime}=I$.

Proposition 3.2**.**

Let $C\in GL^{+}(\mathcal{H})$. The family $\{\Lambda_{j}:j\in J\}$ is a g-frame if and only if $\{\Lambda_{j}:j\in J\}$ is a $C^{2}-$controlled g-frame.

Proof.

Suppose that $\{\Lambda_{j}:j\in J\}$ is a $C^{2}-$controlled g-frame with bounds $A$ and $B$. Then from (3.13), we have

[TABLE]

Now for any $f\in\mathcal{H}$,

[TABLE]

Hence

[TABLE]

Again for any $f\in\mathcal{H}$,

[TABLE]

From (3.15), (3.16) and Theorem 3.2, we conclude that $\{\Lambda_{j}:j\in J\}$ is a g-frame with bound $A\|C\|^{-2}$ and $B\|C^{-1}\|^{2}$.

Conversely, let $\{\Lambda_{j}:j\in J\}$ is a g-frame with bounds $A^{\prime}$ and $B^{\prime}$. Then for all $f\in\mathcal{H}$,

[TABLE]

So for $f\in\mathcal{H}$ we have $Cf\in\mathcal{H}$, and

[TABLE]

Also for any $f\in\mathcal{H}$,

[TABLE]

From (3.17) and (3.18), we have

[TABLE]

Hence $\{\Lambda_{j}:j\in J\}$ is a $C^{2}-$controlled g-frame with bounds $A^{\prime}\|C^{-1}\|^{-2}$ and $B^{\prime}\|C\|^{2}$. ∎

Next, we study when a $g$-frame becomes a $(C,C^{\prime})-$controlled $g$-frame.

Proposition 3.3**.**

Assume that $\{\Lambda_{j}:j\in J\}$ is a g-frame for the Hilbert $C^{*}$-module $\mathcal{H}$ with respect to $\{\mathcal{H}_{j}\}_{j\in J}$. Let $S_{\Lambda}$ be the g-frame operator associated with the g-frame $\{\Lambda_{j}:j\in J\}$ as defined in (2.2). Let $C,C^{\prime}\in GL^{+}(\mathcal{H})$. Then $\{\Lambda_{j}:j\in J\}$ is a $(C,C^{\prime})-$controlled g-frame.

Proof.

$\{\Lambda_{j}:j\in J\}$ is a g-frame for the Hilbert $C^{*}$-module $\mathcal{H}$ with bounds $A$ and $B$. By the equivalence condition (3.14) of g-frame, we have

[TABLE]

Again we have

[TABLE]

and

[TABLE]

From (3.19) and (3.20), we have

[TABLE]

By using Theorem 3.1, we conclude that $\{\Lambda_{j}:j\in J\}$ is a $(C,C^{\prime})-$controlled g-frame with bounds $A\|C\|\|C^{\prime}\|$ and $B\|C\|\|C^{\prime}\|$. ∎

Theorem 3.3**.**

Let $C,C^{\prime}\in GL^{+}(\mathcal{H})$, $\{\Lambda_{j}:j\in J\}\subset End_{A}^{*}(\mathcal{H},\mathcal{H}_{j})$, and $C,C^{\prime}$ commute with each other and commute with $\Lambda_{j}^{*}\Lambda_{j}$ for all $j\in J$. Then the sequence $\{\Lambda_{j}:j\in J\}$ is a $(C,C^{\prime})-$controlled $g$-Bessel sequence for $\mathcal{H}$ with respect to $\{\mathcal{H}_{j}\}_{j\in J}$ with bound $B$ if and only if the operator $T_{(C,C^{\prime})}:l^{2}(\{\mathcal{H}_{j}\}_{j\in J})\rightarrow\mathcal{H}$ given by

[TABLE]

is well defined and bounded operator with $\|T_{(C,C^{\prime})}\|\leq\sqrt{B}$.

Proof.

Let $\{\Lambda_{j}:j\in J\}$ be a $C,C^{\prime}-$controlled g-Bessel sequence for $\mathcal{H}$ with respect to $\{\mathcal{H}_{j}\}_{j\in J}$ with bound $B$. As a result of Theorem 3.1,

[TABLE]

For any sequence $\{g_{j}\}_{j\in J}\in l^{2}(\{\mathcal{H}_{j}\}_{j\in J})$,

[TABLE]

Therefore, the sum $\displaystyle\sum_{j\in J}(CC^{\prime})^{\frac{1}{2}}\Lambda_{j}^{*}g_{j}$ is convergent, and we have

[TABLE]

Hence the operator $T_{(C,C^{\prime})}$ is well defined, bounded and $\|T_{(C,C^{\prime})}\|\leq\sqrt{B}$.

Conversely, let the operator $T_{(C,C^{\prime})}$ is well defined, bounded and $\|T_{(C,C^{\prime})}\|\leq\sqrt{B}$.

For any $f\in\mathcal{H}$ and finite subset $K\subset J$, we have

[TABLE]

where

[TABLE]

Therefore,

[TABLE]

Since $K$ is arbitrary, we have

[TABLE]

Hence we conclude that $\{\Lambda_{j}:j\in J\}$ is a $(C,C^{\prime})-$controlled g-Bessel sequence for $\mathcal{H}$ with respect to $\{\mathcal{H}_{j}\}_{j\in J}$. ∎

Now we prove some perturbation results for $(C,C^{\prime})-$controlled g-frame.

Theorem 3.4**.**

Let $\{\Lambda_{j}\in End_{A}^{*}(\mathcal{H},\mathcal{H}_{j}):j\in J\}$ be a $(C,C^{\prime})-$controlled g-frame for $\mathcal{H}$ with respect to $\{\mathcal{H}_{j}\}_{j\in J}$. Let $\{\Pi_{j}\in End_{A}^{*}(\mathcal{H},\mathcal{H}_{j}):j\in J\}$ be any sequence, and assume that $C$ and $C^{\prime}$ commute with each other and commute with $(\Lambda_{j}-\Pi_{j})^{*}(\Lambda_{j}-\Pi_{j})$. Then $\{\Pi_{j}:j\in J\}$ is a $(C,C^{\prime})-$controlled g-frame for $\mathcal{H}$ with respect to $\{\mathcal{H}_{j}\}_{j\in J}$ if and only if there exists constants $M_{1}$ and $M_{2}$ such that

[TABLE]

Proof.

Let $\{\Lambda_{j}:j\in J\}$ be a $(C,C^{\prime})-$controlled g-frame for $\mathcal{H}$ with lower and upper bounds $A_{1}$ and $B_{1}$, respectively. Also suppose that $\{\Pi_{j}:j\in J\}$ be a $(C,C^{\prime})-$controlled g-frame for $\mathcal{H}$ with lower and upper bounds $A_{2}$ and $B_{2}$, respectively. Then

[TABLE]

Thus (3.22) is proved, where M_{1}=\Big{(}1+\frac{B_{2}}{A_{1}}\Big{)}. In a similar manner, one can obtain

[TABLE]

Hence (3.23) follows with M_{2}=\Big{(}1+\frac{B_{1}}{A_{2}}\Big{)}.

Conversely, suppose that $\{\Lambda_{j}:j\in J\}$ be a $(C,C^{\prime})-$controlled g-frame for $\mathcal{H}$ with lower and upper bounds $A_{1}$ and $B_{1}$, respectively, and (3.22) and (3.23) hold true. Then for any $f\in\mathcal{H}$, using (3.23) we get

[TABLE]

This implies that

[TABLE]

Also we have

[TABLE]

Therefore from (3.24) and (3.25), it is clear that $\{\Pi_{j}:j\in J\}$ is a $(C,C^{\prime})-$controlled g-frame for $\mathcal{H}$ with respect to $\{\mathcal{H}_{j}\}_{j\in J}$. ∎

Proposition 3.4**.**

Let $\{\Lambda_{j}\in End_{A}^{*}(\mathcal{H},\mathcal{H}_{j}):j\in J\}$ and $\{\Gamma_{j}\in End_{A}^{*}(\mathcal{H},\mathcal{H}_{j}):j\in J\}$ be two $(C,C^{\prime})$-controlled $g$-Bessel sequences for $\mathcal{H}$ with respect to $\{\mathcal{H}\}_{j\in J}$ with bounds $B_{1}$ and $B_{2}$, respectively. Then the operator $L_{(C,C^{\prime})}:\mathcal{H}\rightarrow\mathcal{H}$ given by

[TABLE]

is well defined and bounded with $\|L_{(C,C^{\prime})}\|\leq\sqrt{B_{1}B_{2}}$. Also its adjoint operator is $L_{(C,C^{\prime})}^{*}(g)=\displaystyle\sum_{j\in J}C\Lambda_{j}^{*}\Gamma_{j}C^{\prime}g$.

Proof.

For any $f\in\mathcal{H}$ and $K\subset J$, we have

[TABLE]

Since $K$ is arbitrary the series $\displaystyle\sum_{j\in J}C^{\prime}\Gamma_{j}^{*}\Lambda_{j}Cf$ converges in $\mathcal{H}$, and

[TABLE]

Moreover, we see that

[TABLE]

Thus $L_{(C,C^{\prime})}^{*}(g)=\displaystyle\sum_{j\in J}C\Lambda_{j}^{*}\Gamma_{j}C^{\prime}g$. ∎

Theorem 3.5**.**

Let $\{\Lambda_{j}\in End_{A}^{*}(\mathcal{H},\mathcal{H}_{j}):j\in J\}$ be a $(C,C^{\prime})$-controlled $g$-frame for $\mathcal{H}$ with respect to $\{\mathcal{H}\}_{j\in J}$, and $\{\Gamma_{j}\in End_{A}^{*}(\mathcal{H},\mathcal{H}_{j}):j\in J\}$ be a $(C,C^{\prime})$-controlled $g$-Bessel sequence for $\mathcal{H}$ with respect to $\{\mathcal{H}\}_{j\in J}$. Assume that $C$ and $C^{\prime}$ commute with each other and commute with $\Gamma_{j}^{*}\Gamma_{j}$. If the operator $L_{(C,C^{\prime})}$ defined in (3.26) is surjective then $\{\Gamma_{j}:j\in J\}$ is also a $(C,C^{\prime})$-controlled $g$-frame for $\mathcal{H}$ with respect to $\{\mathcal{H}\}_{j\in J}$.

Proof.

It is given that $\{\Lambda_{j}:j\in J\}$ is a $(C,C^{\prime})$-controlled $g$-frame for $\mathcal{H}$ with respect to $\{\mathcal{H}\}_{j\in J}$. Then by Theorem 3.3, the operator $T_{(C,C^{\prime})}:l^{2}(\{\mathcal{H}_{j}\}_{j\in J})\rightarrow\mathcal{H}$ given by

[TABLE]

is well defined and bounded operator. By (3.6) its adjoint operator $T^{*}_{(C,C^{\prime})}:\mathcal{H}\rightarrow l^{2}(\{\mathcal{H}_{j}\}_{j\in J})$ is given by

[TABLE]

Since $\{\Gamma_{j}:j\in J\}$ is also a $(C,C^{\prime})$-controlled $g$-Bessel sequence for $\mathcal{H}$ with respect to $\{\mathcal{H}\}_{j\in J}$, again by Theorem 3.3, the operator $P_{(C,C^{\prime})}:l^{2}(\{\mathcal{H}_{j}\}_{j\in J})\rightarrow\mathcal{H}$ given by

[TABLE]

is well defined and bounded operator. Again its adjoint operator is given by

[TABLE]

Hence for any $f\in\mathcal{H}$, the operator defined in (3.26) can be written as

[TABLE]

Since $L_{(C,C^{\prime})}$ is surjective then for any $f\in\mathcal{H}$, there exists $g\in\mathcal{H}$ such that $f=L_{(C,C^{\prime})}(g)=P_{(C,C^{\prime})}T^{*}_{(C,C^{\prime})}(g)$, and $T^{*}_{(C,C^{\prime})}(g)\in l^{2}(\{\mathcal{H}_{j}\}_{j\in J})$. This implies that $P_{(C,C^{\prime})}$ is surjective. As a result of Lemma 3.1, we have $P^{*}_{(C,C^{\prime})}$ is bounded below, that is there exists $m>0$ such that

[TABLE]

Hence $\{\Gamma_{j}:j\in J\}$ is also a $(C,C^{\prime})$-controlled $g$-frame for $\mathcal{H}$ with respect to $\{\mathcal{H}\}_{j\in J}$. ∎