On Controlled Frames in Hilbert $C^*$-modules

TL;DR

This paper introduces controlled frames in Hilbert C*-modules, explores their properties, characterizations, and related operators, and investigates their connections to weighted frames within this mathematical framework.

Contribution

It extends the concept of controlled frames to Hilbert C*-modules, providing new characterizations and analyzing associated multiplier and weighted frames.

Findings

Controlled frames share key properties with those in Hilbert spaces.

Characterization of controlled frames in Hilbert C*-modules.

Development of multiplier operators for controlled frames.

Abstract

In this paper, we introduce controlled frames in Hilbert $C^*$-modules and we show that they share many useful properties with their corresponding notions in Hilbert space. Next, we give a characterization of controlled frames in Hilbert $C^*$-module. Also multiplier operators for controlled frames in Hilbert $C^*$-modules will be defined and some of its properties will be shown. Finally, we investigate weighted frames in Hilbert $C^*$-modules and verify their relations to controlled frames and multiplier operators.

Equations

$$m\Vert f{\Vert }^2\le \sum _j⟨f,{\psi }_j⟩⟨C{\psi }_j,f⟩\le M\Vert f{\Vert }^2,$$

$$m\Vert f{\Vert }^2\le \sum _j{\omega }_j|⟨f,{\psi }_j⟩{|}^2\le M\Vert f{\Vert }^2,$$

$${\ell }^2(A)=\left\{\{a_j\}\subseteq A:\sum _{j\in J}{a}_j^{\ast }{a}_{j}convergesin\Vert .\Vert \right\}$$

$$⟨\{a_j\},\{b_j\}⟩=\sum _{j\in J}{a}_j^{\ast }{b}_j,\{a_j\},\{b_j\}\in {\ell }^2(A)$$

$$\Vert \{a_j\}\Vert :=\sqrt{\Vert \sum a_j^{\ast }{a}_j\Vert },$$

$$\Vert ⟨f,g⟩{\Vert }^2\le \Vert ⟨f,f⟩\Vert \times \Vert ⟨g,g⟩\Vert .$$

$$C⟨f,f⟩\le \sum _{j\in J}⟨f,f_j⟩⟨f_j,f⟩\le D⟨f,f⟩$$

$$C\Vert f{\Vert }^2\le \Vert \sum _{j\in J}⟨f,f_j⟩⟨f_j,f⟩\Vert \le D\Vert f{\Vert }^2,f\in \mathcal{H}.$$

$$f=\sum _{j\in J}⟨f,g_j⟩f_j$$

$$Sf=\sum _{j\in J}⟨f,f_j⟩f_j,f\in \mathcal{H}$$

$$⟨Sf,f⟩=⟨\sum _{j\in J}⟨f,f_j⟩f_j,f⟩=\sum _{j\in J}⟨f,f_j⟩⟨f_j,f⟩,f\in \mathcal{H}$$

$$C⟨f,f⟩\le ⟨Sf,f⟩\le D⟨f,f⟩,f\in \mathcal{H}$$

$$f=S{S}^{-1}f=S^{-1}Sf=\sum _{j\in J}⟨S^{-1}f,f_j⟩f_j=\sum _{j\in J}⟨f,S^{-1}{f}_j⟩f_j$$

$$f=\sum _{j\in J}⟨f,\widetilde{f_j}⟩f_j=\sum _{j\in J}⟨f,f_j⟩\widetilde{f_j},$$

$${\mathbf{M}}_{m,({\varphi }_j),({\psi }_j)}:{\mathcal{H}}_1\to {\mathcal{H}}_2$$

$${\mathbf{M}}_{m,({\varphi }_j),({\psi }_j)}(f)=\sum _{j\in J}{m}_j⟨f,{\psi }_j⟩{\varphi }_j$$

$$M_{m,\{f_j\},\{g_j\}}f:=\sum _{j\in J}{m}_j⟨f,f_j⟩g_j,f\in \mathcal{H}$$

$$m⟨f,f⟩\le \sum _{j\in J}⟨f,{\psi }_j⟩⟨C{\psi }_j,f⟩\le M⟨f,f⟩,$$

$$\sum _{j\in J}⟨f,{\psi }_j⟩⟨C{\psi }_j,f⟩\le M⟨f,f⟩,$$

$$S_{C}f=\sum _{j\in J}⟨f,{\psi }_j⟩C{\psi }_j$$

$$\sum _{j\in J}⟨f,{\psi }_j⟩⟨C{\psi }_j,f⟩\le M⟨f,f⟩,$$

$$S_{C}f=\sum _{j\in J}⟨f,{\psi }_j⟩C{\psi }_j$$

$$⟨S_{C}f,g⟩$$

$$=\sum _{j\in J}⟨f,{\psi }_j⟩⟨{\psi }_j,C^{\ast }g⟩=⟨Sf,C^{\ast }g⟩=⟨f,S{C}^{\ast }g⟩.$$

$$⟨S_{C}f,f⟩=⟨\sum _{j\in J}⟨f,{\psi }_j⟩C{\psi }_j,f⟩=\sum _{j\in J}⟨f,{\psi }_j⟩⟨C{\psi }_j,f⟩.$$

$$m⟨f,f⟩\le ⟨S_{C}f,f⟩\le M⟨f,f⟩,$$

$$mI{d}_{\mathcal{H}}\le S_C\le MI{d}_{\mathcal{H}}.$$

$$m\Vert f{\Vert }^2\le \Vert \sum _{j\in J}⟨f,{\psi }_j⟩⟨C{\psi }_j,f⟩\Vert \le M\Vert f{\Vert }^2,$$

$$C\Vert f{\Vert }^2\le \Vert \sum _{j\in J}⟨f,{\psi }_j⟩⟨C{\psi }_j,f⟩\Vert \le D\Vert f{\Vert }^2,$$

$$\Vert \sum _{j\in J}⟨f,{\psi }_j⟩⟨{\psi }_j,f⟩\Vert =\Vert ⟨Sf,f⟩\Vert \le \Vert Sf\Vert \Vert f{\Vert }^2\le D\Vert f{\Vert }^2.$$

$$\Vert ⟨Sf,f⟩\Vert \ge \Vert ⟨Cf,f⟩\Vert =C\Vert f{\Vert }^2.$$

Full text

1. Introduction

Frames for Hilbert spaces were first introduced in 1952 by Duffin and Schaeffer [11] for study of nonharmonic Fourier series. They were reintroduced and development in 1986 by Daubechies, Grossmann and Meyer[10], and popularized from then on. For basic results on frames, see [8].

Hilbert $C^{*}$-modules form a wide category between Hilbert spaces and Banach spaces. Their structure was first used by Kaplansky [20] in 1952. They are an often used tool in operator theory and in operator algebra theory. They serve as a major class of examples in operator $C^{*}$-module theory.

The notions of frames in Hilbert $C^{*}$-modules were introduced and investigated in [14]. Frank and Larson [14, 15] defined the standard frames in Hilbert $C^{*}$-modules in 1998 and got a series of result for standard frames in finitely or countably generated Hilbert $C^{*}$-modules over unital $C^{*}$-algebras. Extending the results to this more general framework is not a routine generalization, as there are essential differences between Hilbert $C^{*}$-modules and Hilbert spaces. For example, any closed subspace in a Hilbert space has an orthogonal complement, but this fails in Hilbert $C^{*}$-module. Also there is no explicit analogue of the Riesz representation theorem of continuous functionals in Hilbert $C^{*}$-modules. We refer the readers to [24] and [19] for more details on Hilbert $C^{*}$-modules and to [15, 28, 30, 29] for a discussion of basic properties of frame in Hilbert $C^{*}$-modules and their generalizations.

Controlled frames have been introduced to improve the numerical efficiency of iterative algorithms for inverting the frame operator on abstract Hilbert spaces [5], however they are used earlier in [7] for spherical wavelets.

In this paper, we introduce controlled frames in Hilbert $C^{*}$-modules and we show that they share many useful properties with their corresponding notions in Hilbert space. Next, we give a characterization of controlled frames in Hilbert $C^{*}$-module. Also multiplier operators for controlled frames in Hilbert $C^{*}$-modules will be defined and some of its properties will be shown. Finally, we investigate weighted frames in Hilbert $C^{*}$-modules and verify their relation to controlled frames and multiplier operators.

The paper is organized as follows. In section 2, we review the concept Hilbert $C^{*}$-modules, frames and multiplier operators in Hilbert $C^{*}$-modules. Also the analysis, synthesis, frame operator and dual frames be reviewed. In section 3, we introduce controlled frames in Hilbert $C^{*}$-modules and characterize them. In section 4, we investigate weighted frames in Hilbert $C^{*}$-modules and verify their relation to controlled frames and multiplier operators.

2. Preliminaries

In this section, we collect the basic notations and some preliminary results. We denote by $I$ the identity operator on $\mathcal{H}$. Let $B(\mathcal{H}_{1},\mathcal{H}_{2})$ be the set of all bounded linear operators from $\mathcal{H}_{1}$ to $\mathcal{H}_{2}$. This set is a Banach space for the operator norm $\|A\|=sup_{\|x\|_{\mathcal{H}_{1}}\leq 1}\|Ax\|_{\mathcal{H}_{2}}$. The adjoint of the operator $A$ is denoted by $A^{*}$ and the spectrum of $A$ by $\sigma(A)$. We define $GL(\mathcal{H}_{1},\mathcal{H}_{2})$ as the set of all bounded linear operators with a bounded inverse, and similarly for $GL(\mathcal{H})$. Our standard reference for Hilbert space and operator theory is [9].

Controlled frames introduced in [5] as follows. Let $C\in GL(\mathcal{H})$. A frame controlled by the operator $C$ or $C$-controlled frame is a family of vectors $\Psi=\{\psi_{j}\in\mathcal{H}:j\in J\}$, such that there exist two constants $m>0$ and $M<\infty$ satisfying

[TABLE]

for all $f\in\mathcal{H}$.

Also weighted frames are defined as follows. Let $\Psi=\{\psi_{j}\in\mathcal{H}:j\in J\}$ be a sequence of elements in $\mathcal{H}$ and $\{\omega_{j}:j\in J\}\subseteq\mathbb{R}^{+}$a sequence of positive weights. This pair is called a weighted frame of $\mathcal{H}$ if there exist constants $m>0$ and $M<\infty$ such that

[TABLE]

for all $f\in\mathcal{H}$.

Hilbert $C^{*}$-modules form a wide category between Hilbert spaces and Banach spaces. Hilbert $C^{*}$-modules are generalizations of Hilbert spaces by allowing the inner product to take values in a $C^{*}$-algebra rather than in the field of complex numbers.

Let $A$ be a $C^{*}$-algebra with involution $*$. An inner product $A$-module (or pre Hilbert $A$-module) is a complex linear space $\mathcal{H}$ which is a left $A$-module with an inner product map $\langle.,.\rangle:\mathcal{H}\times{\mathcal{H}}\rightarrow A$ which satisfies the following properties:

1. (1)

$\langle\alpha{f}+\beta{g},h\rangle=\alpha\langle f,h\rangle+\beta\langle g,h\rangle$ for all $f,g,h\in\mathcal{H}$ and $\alpha,\beta\in\mathbb{C}$; 2. (2)

$\langle af,g\rangle=a\langle f,g\rangle$ for all $f,g\in\mathcal{H}$ and $a\in A$; 3. (3)

$\langle f,g\rangle=\langle g,f\rangle^{*}$ for all $f,g\in\mathcal{H}$; 4. (4)

$\langle f,f\rangle\geq 0$ for all $f\in\mathcal{H}$ and $\langle f,f\rangle=0$ iff $f=0$.

For $f\in\mathcal{H}$, we define a norm on $\mathcal{H}$ by $\|f\|_{\mathcal{H}}=\|\langle{f},{f}\rangle\|^{{1}/{2}}_{A}$. If $\mathcal{H}$ is complete with this norm, it is called a (left) Hilbert $C^{*}$-module over $A$ or a (left) Hilbert $A$-module.

An element $a$ of a $C^{*}$-algebra $A$ is positive if $a^{*}=a$ and its spectrum is a subset of positive real numbers. In this case, we write $a\geq 0$. By condition (4) in the definition $\langle{f},{f}\rangle\geq 0$ for every $f\in\mathcal{H}$, hence we define $|f|=\langle{f},{f}\rangle^{1/2}$. We call $Z(A)=\{a\in A:ab=ba,\forall b\in A\}$, the center of $A$. If $a\in Z(A)$, then $a^{*}\in Z(A)$, and if $a$ is an invertible element of $Z(A)$, then $a^{-1}\in Z(A)$, also if $a$ is a positive element of $Z(A)$, then $a^{\frac{1}{2}}\in Z(A)$. Let $Hom_{A}(M,N)$ denotes the set of all $A$-linear operators from $M$ to $N$.

Let

[TABLE]

with inner product

[TABLE]

and

[TABLE]

it was shown that [33], $\ell^{2}(A)$ is Hilbert $A$-module.

Note that in Hilbert $C^{*}$-modules the Cauchy-Schwartz inequality is valid.

Let $f,g\in\mathcal{H}$, where $\mathcal{H}$ is a Hilbert $C^{*}$-module , then

[TABLE]

We are focusing in finitely and countably generated Hilbert $C^{*}$- modules over unital $C^{*}$-algebra $A$. A Hilbert $A$-module $\mathcal{H}$ is finitely generated if there exists a finite set $\{x_{1},x_{2},...,x_{n}\}\subseteq\mathcal{H}$ such that every $x\in\mathcal{H}$ can be expressed as $x=\sum_{i=1}^{n}a_{i}x_{i}$, $a_{i}\in A$. A Hilbert $A$-module $\mathcal{H}$ is countably generated if there exits a countable set of generators.

The notion of (standard) frames in Hilbert $C^{*}$-modules is first defined by Frank and Larson [15]. Basic properties of frames in Hilbert $C^{*}$-modules are discussed in [16, 17, 21, 22].

Let $\mathcal{H}$ be a Hilbert $C^{*}$-module, and $J$ a set which is finite or countable. A system $\{f_{j}:j\in J\}\subseteq\mathcal{H}$ is called a frame for $\mathcal{H}$ if there exist constants $C,D>0$ such that

[TABLE]

for all $f\in\mathcal{H}$. The constants $C$ and $D$ are called the frame bounds. If $C=D$ it called a tight frame and in the case $C=D=1$, it called Parseval frame. It is called a Bessel sequence if the second inequality in (2.1) holds.

Unlike Banach spaces, it is known [15] that every finitely generated or countably generated Hilbert $C^{*}$- modules admits a frame.

The following characterization of frames in Hilbert $C^{*}$- modules, which was obtained independently in [1] and [18], enables us to verify whether a sequence is a frames in Hilbert $C^{*}$- modules in terms of norms. It also allows us to characterize frames in Hilbert $C^{*}$- modules from the operator theory point of view.

Theorem 2.1**.**

Let $\mathcal{H}$ be a finitely or countably generated Hilbert $A$-module over a unital $C^{*}$-algebra $A$ and $\{f_{j}:j\in J\}\subseteq\mathcal{H}$ a sequence. Then $\{f_{j}:j\in J\}$ is a frame for $\mathcal{H}$ if and only if there exist constants $C,D>0$ such that

[TABLE]

Let $\{f_{j}:j\in J\}$ be a frame in Hilbert $A$-module $\mathcal{H}$ and $\{g_{j}:j\in J\}$ be a sequence of $\mathcal{H}$. Then $\{g_{j}:j\in J\}$ is called a dual sequence of $\{f_{j}:j\in J\}$ if

[TABLE]

for all $f\in\mathcal{H}$. The sequences $\{f_{j}:j\in J\}$ and $\{g_{j}:j\in J\}$ are called a dual frame pair when $\{g_{j}:j\in J\}$ is also a frame.

For the frame $\{f_{j}:j\in J\}$ in Hilbert $A$-module $\mathcal{H}$, the operator $S$ defined by

[TABLE]

is called the frame operator. It is proved that [15], $S$ is invertible, positive, adjointable and self-adjoint. Since

[TABLE]

it follows that

[TABLE]

and the following reconstruction formula holds

[TABLE]

for all $f\in\mathcal{H}$.

Let $\tilde{f}_{j}=S^{-1}f_{j}$, then

[TABLE]

for any $f\in\mathcal{H}$. The sequence $\{\tilde{f}_{j}:j\in J\}$ is also a frame for $\mathcal{H}$ which is called the canonical dual frame of $\{f_{j}:j\in J\}$.

In [31], R. Schatten provided a detailed study of ideals of compact operators using their singular decomposition. He investigated the operators of the form $\sum_{j}\lambda_{j}\varphi_{j}\otimes\psi_{j}$ where $(\phi_{j})$ and $(\psi_{j})$ are orthonormal families. In [4], the orthonormal families were replaced with Bessel and frame sequences to define Bessel and frame multipliers.

Definition 2.2**.**

Let $\mathcal{\mathcal{H}}_{1}$ and $\mathcal{\mathcal{H}}_{2}$ be Hilbert spaces, let $(\psi_{j})\subseteq\mathcal{\mathcal{H}}_{1}$ and $(\phi_{j})\subseteq\mathcal{H}_{2}$ be Bessel sequences. Fix $m=(m_{j})\in l^{\infty}$. The operator

[TABLE]

defined by

[TABLE]

for all $f\in\mathcal{H}_{1}$ is called the Bessel multiplier for the Bessel sequences $(\psi_{j})$ and $(\phi_{j})$. The sequence $m$ is called the symbol of M. For frames, this operator is called frame multiplier, for Riesz sequences a Riesz multiplier.

Basic properties and some applications of this operator for Bessel sequences, frames and Riesz basis have been proved by Peter Balazs in his Ph.D habilation [3]. Recently, the concept of multipliers extended and introduced for continuous frames [6], fusion frames [2], $p$-Bessel sequences [26], generalized frames [25], controlled frames [27], Banach frames [12, 13], Hilbert $C^{*}$-modules [23] and etc.

Definition 2.3**.**

Let $A$ be a unital $C^{*}$-algebra, $J$ be a finite or countable index set and $\{f_{j}:j\in J\}$ and $\{g_{j}:j\in J\}$ be Hilbert $C^{*}$-modules Bessel sequences for $\mathcal{H}$. For $m=\{m_{j}\}_{j\in J}\in\ell^{\infty}(A)$ with $m_{j}\in Z(A)$, for each $j\in J$, the operator $M_{m,\{f_{j}\},\{g_{j}\}}:\mathcal{H}\to\mathcal{H}$ defined by

[TABLE]

called the multiplier operator of $\{f_{j}:j\in J\}$ and $\{g_{j}:j\in J\}$. The sequence $m=\{m_{j}\}_{j}\in J$ called the symbol of $M_{m,\{f_{j}\},\{g_{j}\}}$.

The symbol of $m$ has important role in the studying of multiplier operators. In this paper $m$ is always a sequence $m=\{m_{j}\}_{j\in J}\in\ell^{\infty}(A)$ with $m_{j}\in Z(A)$, for each $j\in J$.

3. Controlled Frames In Hilbert $C^{*}$-modules

In this section, we introduce controlled frames in Hilbert $C^{*}$-modules and we show that they share many useful properties with their corresponding notions in Hilbert space. We also give a characterization of controlled frames in Hilbert $C^{*}$-module.

Definition 3.1**.**

Let $\mathcal{H}$ be a Hilbert $C^{*}$-module and $C\in GL(\mathcal{H})$. A frame controlled by the operator $C$ or $C$-controlled frame in Hilbert C*-module $\mathcal{H}$ is a family of vectors $\Psi=\{\psi_{j}\in\mathcal{H}:j\in J\}$, such that there exist two constants $m>0$ and $M<\infty$ satisfying

[TABLE]

for all $f\in\mathcal{H}$.

Likewise, $\Psi=\{\psi_{j}\in\mathcal{H}:j\in J\}$ is called a $C$-controlled Bessel sequence with bound $M$ if there exists $M<\infty$ such that

[TABLE]

for every $f\in\mathcal{H}$, where the sum in the inequality is convergent in norm.

If $m=M$, we call this $C$-controlled frame a tight $C$-controlled frame, and if $m=M=1$ it is called a Parseval $C$-controlled frame.

Every frame is a $I$-controlled frame. Hence controlled frames are generalizations of frames.

The proof of the following lemma is straightforward.

Lemma 3.2**.**

Let $\mathcal{H}$ be a Hilbert $C^{*}$-module and $C\in GL(\mathcal{H})$. A sequence $\Psi=\{\psi_{j}\in\mathcal{H}:j\in J\}$ is $C$-controlled Bessel sequence in Hilbert C-module $\mathcal{H}$ if and only if the operator*

[TABLE]

is well defined and there exists constant $M<\infty$ such that

[TABLE]

for every $f\in\mathcal{H}$.

By using Lemma 3.2 we have the following definition.

Definition 3.3**.**

Let $\mathcal{H}$ be a Hilbert $C^{*}$-module and $C\in GL(\mathcal{H})$. Assume the sequence $\Psi=\{\psi_{j}\in\mathcal{H}:j\in J\}$ is $C$-controlled Bessel sequence in Hilbert C*-module $\mathcal{H}$. The operator

[TABLE]

is called $C$-controlled frame operator.

According to the following proposition, the main properties of controlled frame operators in Hilbert $C^{*}$-modules are the same as controlled frame operators in Hilbert spaces.

Proposition 3.4**.**

Let $\mathcal{H}$ be a Hilbert $C^{*}$-module on $C^{*}$-algebra $A$ and $C\in End_{A}^{*}(U,V)$. Assume $\{\psi_{j}:j\in J\}$ is a C-controlled frame in Hilbert $C^{*}$-module $\mathcal{H}$ with bounds $m,M>0$ Then $C$-controlled frame operator $S_{C}$ is invertible, positive, adjointable and self-adjoint.

Proof.

Let $f,g\in\mathcal{H}$ and $Sf=\sum_{j\in J}\langle f,\psi_{j}\rangle\psi_{j}$ be frame operator of $\{\psi_{j}:j\in J\}$, then

[TABLE]

Therefore controlled frame operator $S_{C}$ is adjointable and $S_{C}^{*}=SC^{*}$.

Since

[TABLE]

It follows that

[TABLE]

and

[TABLE]

So $S_{C}$ is positive and invertible. By regard to this fact that every positive operator in Banach space is self-adjoint, $C$-controlled frame operator $S_{C}$ is self-adjoint. ∎

Now, by using the following lemma in [1], we give a characterization of controlled frames.

Lemma 3.5**.**

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

1. (1)

T is surjective; 2. (2)

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

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

Theorem 3.6**.**

Let $\mathcal{H}$ be a Hilbert $C^{*}$-module and $C\in GL(\mathcal{H})$. A sequence $\Psi=\{\psi_{j}\in\mathcal{H}:j\in J\}$ is $C$-controlled frame in Hilbert $C^{*}$-module $\mathcal{H}$ if and only if there exists constants $m>0$ and $M<\infty$ such that

[TABLE]

for all $f\in\mathcal{H}$.

Proof.

Let the sequence $\Psi=\{\psi_{j}\in\mathcal{H}:j\in J\}$ is $C$-controlled frame in Hilbert $C^{*}$-module $\mathcal{H}$. By the definition of $C$-controlled frame inequality (3.1) holds.

Now suppose that the inequality (3.1) holds. From Proposition 3.4 $C$-controlled frame operator $S_{C}$ is positive, self-adjoint and invertible, hence $\langle S_{C}^{\frac{1}{2}}f,S_{C}^{\frac{1}{2}}f\rangle=\langle S_{C}f,f\rangle=\sum_{j\in J}\langle f,\psi_{j}\rangle\langle C\psi_{j},f\rangle$. So we have $\sqrt{m}\|f\|\leq\|S_{C}^{\frac{1}{2}}f\|\leq\sqrt{M}\|f\|$ for any $f\in\mathcal{H}$. According to Lemma 3.5 there are constants $C,D>0$ such that

[TABLE]

which implies that $\Psi=\{\psi_{j}\in\mathcal{H}:j\in J\}$ is $C$-controlled frame in Hilbert $C^{*}$-module $\mathcal{H}$. ∎

We prove the following theorems to show that every controlled frame is a classical frame.

Theorem 3.7**.**

Let $\mathcal{H}$ be a Hilbert $C^{*}$-module on $C^{*}$-algebra $A$ and $\{\psi_{j}\}_{j\in J}$ be a sequence in $\mathcal{H}$. Assume $S$ is the frame operator associated with $\{\psi_{j}\}_{j\in J}$ i.e. $Sf=\sum_{j\in J}\langle f,\psi_{j}\rangle\psi_{j}$. Then the following conditions are equivalent:

1. (1)

$\{\psi_{j}\}_{j\in J}$* is a frame with frame bounds $C$ and $D$;* 2. (2)

We have $CId_{\mathcal{H}}\leq S\leq DId_{\mathcal{H}}$.

Proof.

$(1)\Rightarrow(2).$ This is implied by proof of Proposition 3.4.

$(2)\Rightarrow(1).$ Let $T$ be the analysis operator associated with $\{\psi_{j}\}_{j\in J}$. Since $S=T^{*}T$ and hence $\|T\|^{2}=\|S\|$ for any $f\in\mathcal{H}$ we obtain

[TABLE]

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

[TABLE]

Therefor for all $f\in\mathcal{H}$

[TABLE]

Now, by Theorem 2.1 proof is complete. ∎

Theorem 3.8**.**

[8]** Let $X$ be a Banach space, $U:X\rightarrow X$ a bounded operator and $\|I-U\|<1$. Then $U$ is invertible.

Theorem 3.9**.**

Let $T:\mathcal{H}\rightarrow\mathcal{H}$ be a linear operator. Then the following conditions are equivalent:

1. (1)

There exist $m>0$ and $M<\infty$ such that $mI\leq T\leq MI$; 2. (2)

$T\in GL^{+}(\mathcal{H})$* i.e.$T$ is invertible and positive.*

Proof.

$(1)\Rightarrow(2)$ Since $mI\leq T$, then $0<(T-mI)$, therefore

[TABLE]

. Hence

[TABLE]

therefore $T$ is positive.

Since $mI\leq T\leq MI$, then $0\leq I-M^{-1}T\leq\left(\frac{M-m}{M}\right)$. Therefore $\|I-M^{-1}T\|\leq\left(\frac{M-m}{M}\right)<1$. Now by Theorem 3.8 the operator $T$ is invertible.

$(1)\Rightarrow(2)$ Let $\|T\|=M$. Since $\|Th\|\leq\|T\|\|h\|$ for every $h\in\mathcal{H}$, then $\|Th\|\leq M\|h\|$ for every $h\in\mathcal{H}$, therefore $T<MI$. Also,

[TABLE]

hence

[TABLE]

for every $h\in\mathcal{H}$. Suppose that $m=\|T^{-1}\|^{-1}$. Since $T$ is positive, $mI\leq T$. ∎

The following proposition shows that any $C$-controlled frame is a controlled frame.

Proposition 3.10**.**

Let sequence $\Psi=\{\psi_{j}\in\mathcal{H}:j\in J\}$ be $C$-controlled frame in Hilbert C-module $\mathcal{H}$ for $C\in GL(\mathcal{H})$. Then $\Psi$ is a frame in Hilbert C*-module $\mathcal{H}$. Furthermore $CS=SC^{*}$ and so*

[TABLE]

Proof.

Define $S:=C^{-1}S_{C}$. Then for every $f\in\mathcal{H}$

[TABLE]

The operator $S:\mathcal{H}\rightarrow\mathcal{H}$ is positive, invertible and self-adjoint. Now by Theorem 3.7 and Theorem 3.9 $\Psi$ is a frame.

Since the operator $S_{C}$ is self-adjoint and $S_{C}=CS$, then $CS=S_{C}=S_{C}^{*}=S^{*}C^{*}=SC^{*}.$ ∎

According to the following proposition, if the operator $C$ is self-adjoint, then $C$-controlled frames are equivalent to classical frames. this is a generalization of Proposition 3.3 in [5] for Hilbert $C^{*}$-module setting.

Proposition 3.11**.**

Let $\mathcal{H}$ be a Hilbert $C^{*}$-module and $C\in GL(\mathcal{H})$ be self-adjoint. Then $\{\psi_{j}\in\mathcal{H}:j\in J\}$ is a $C$- controlled frame for $\mathcal{H}$ if and only if $\Psi$ is a frame for $\mathcal{H}$ and $C$ is positive and commutes with frame operator $S$ .

Proof.

Let $\{\psi_{j}:j\in J\}$ be a $C$-controlled frame for $\mathcal{H}$. Then from Proposition 3.10 $\{\psi_{j}:j\in J\}$ is a frame $\mathcal{H}$ and $C$ is commutes with frame operator $S$. Therefore $C=S_{C}C^{-1}$ is positive.

For the converse, we note that, if $\{\psi_{j}:j\in J\}$ is a frame, then the frame operator $S$ is positive and invertible. Therefore $CS=S_{C}$ is positive and invertible. Now, by Theorem 3.9 $\{\psi_{j}:j\in J\}$ is a $C$-controlled frame for $\mathcal{H}$. ∎

4. Multipliers of controlled frames and weighted frames in Hilbert $C^{*}$-modules

In this section, we generalize the concept of multipliers of frames for controlled frames in Hilbert $C^{*}$-module. Then we investigate weighted frames in Hilbert $C^{*}$-modules and verify their relation to controlled frames and multiplier operators.

Proposition 4.1**.**

Let $\mathcal{H}$ be a Hilbert $C^{*}$-module and $C\in GL(\mathcal{H})$. Assume $\Phi=\{\phi_{j}\in\mathcal{H}:j\in J\}$ and $\Psi=\{\psi_{j}\in\mathcal{H}:j\in J\}$ are $C$-controlled Bessel sequence for $\mathcal{H}$. Then the operator

[TABLE]

defined by

[TABLE]

is a well defined bounded operator.

Proof.

Let $\Phi=\{\phi_{j}\in\mathcal{H}:j\in J\}$ and $\Psi=\{\psi_{j}\in\mathcal{H}:j\in J\}$ be $C$-controlled Bessel sequences for $\mathcal{H}$ with bounds $D$ and $D^{\prime}$, respectively. For any $f,g\in\mathcal{H}$ and finite subset $I\subset J$,

[TABLE]

This show that $M_{m,\Phi,\Psi}$ is well defined and

[TABLE]

∎

Above lemma is a motivation to define the following definition.

Definition 4.2**.**

Let $\mathcal{H}$ be a Hilbert $C^{*}$-module and $C\in GL(\mathcal{H})$. Assume $\Psi=\{\psi_{j}\in\mathcal{H}:j\in J\}$ and $\Phi=\{\phi_{j}\in\mathcal{H}:j\in J\}$ are $C$-controlled Bessel sequences for $\mathcal{H}$. Then the operator

[TABLE]

defined by

[TABLE]

is called the $C$-controlled multiplier operator with symbol $m$.

The following definition is a generalization of weighted frames in Hilbert space to Hilbert $C^{*}$-module.

Definition 4.3**.**

Let $\Psi=\{\psi_{j}\in\mathcal{H}:j\in J\}$ be a sequence of elements in Hilbert $C^{*}$-module $\mathcal{H}$ and $\{\omega_{j}\}_{j\in J}\subseteq Z(A)$ a sequence of positive weights. This pair is called a $w$-frame of Hilbert $C^{*}$-module $\mathcal{H}$ if there exist constants $C,D>0$ and such that

[TABLE]

for all $f\in\mathcal{H}$.

A sequence$\{c_{j}:j\in J\}\in Z(A)$ is called semi-normalized if there are bounds $b\geq a>0$, such that $a\leq|c_{n}|\leq b$.

The following proposition gives a relation between controlled frames, weighted frames and multiplier operators.

Proposition 4.4**.**

Let $\mathcal{H}$ be a Hilbert $C^{*}$-module and $C\in GL(\mathcal{H})$ be self-adjoint and diagonal on $\Psi=\{\psi_{j}\in\mathcal{H}:j\in J\}$ and assume it generates a controlled frame. Then the sequence $W=\{\omega_{j}\}_{j\in J}\subseteq Z(A)$, which verifies the relations $C\psi_{n}=\omega_{n}\psi_{n}$, is semi-normalized and positive. Furthermore $C=M_{W,\tilde{\Psi},\Psi}$.

Proof.

By Theorem 3.9, we get the following result for $C^{1/2}$:

[TABLE]

As $C\psi_{j}=\omega_{j}\psi_{j}$, clearly $C^{1/2}\psi_{j}=\sqrt{\omega_{j}}\psi_{j}$. Applying the above inequalities to the elements of the sequence, we get $0<m\leq\omega_{j}\leq M$.

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

[TABLE]

∎

As a to the first part of Proposition 4.4 a frame weighted by semi-normalized sequence is always a frame. Indeed, we have the following lemma.

Lemma 4.5**.**

Let $\{\omega_{j}:j\in J\}$ be a semi-normalized sequence with bounds $a$ and $b$. If $\{\psi_{j}:j\in J\}$ is a frame with bounds $C$ and $D$ in Hilbert $C*$-module $\mathcal{H}$, then $\{\omega_{j}\psi_{j}:j\in J\}$ is also a frame with bounds $a^{2}C$ and $b^{2}D$. The sequence $\{\omega^{-1}_{j}\tilde{\psi}_{j}:j\in J\}$ is a dual frame of $\{\omega_{j}\psi_{j}:j\in J\}$.

Proof.

Since for any $f\in\mathcal{H}$ $|\langle f,\omega_{j}\psi_{j}\rangle|^{2}=|\omega_{j}|^{2}|\langle f,\psi_{j}\rangle|^{2}$, we get

[TABLE]

Thus $\Delta\leq b^{2}\sum_{j\in J}|\langle f,\omega_{j}\psi_{j}\rangle|^{2}\leq b^{2}D\|f\|^{2}$. In addition,

[TABLE]

As $\sum_{j\in J}\langle f,\omega_{j}\psi_{j}\rangle\omega_{j}^{-1}\tilde{\psi_{j}}=\sum_{j\in J}\langle f.\psi_{j}\rangle\tilde{\psi_{j}}=f$, these two sequences are dual. Since $\omega_{j}^{-1}$ is bounded, $\{\omega_{j}^{-1}\tilde{\psi_{j}}:j\in J\}$ is a Bessel sequence dual to a frame. Therefore, it is a dual frame of $\{\omega_{j}\psi_{j}:j\in J\}$. ∎

The following results give a connection between weighted frames and frame multipliers.

Lemma 4.6**.**

Let $\Psi=\{\psi_{j}\in\mathcal{H}:j\in J\}$ be a frame for Hilbert $C^{*}$-module $\mathcal{H}$. Let $m=\{m_{j}\}_{j\in J}$ be a positive and semi-normalized sequence. Then the multiplier $M_{m,\Psi}$ is the frame operator of the frame $\{\sqrt{m_{j}}\psi_{j}:j\in J\}$ and therefore it is positive, self-adjoint and invertible. If $\{m_{j}\}_{j\in J}$ is negative and semi-normalized, then $M_{m,\Psi}$ is negative, self-adjoint and invertible.

Proof.

[TABLE]

By Lemma 4.5, $\{\sqrt{m_{j}}\psi_{j}:j\in J\}$ is a frame. Therefore $M_{m,\Psi}=S_{(\sqrt{m_{j}}\psi_{j})}$ is positive and invertible.

Let $m_{j}<0$ for all $j\in J$, then $m_{j}=-\sqrt{|m_{j}|^{2}}$. Therefore

[TABLE]

∎

Theorem 4.7**.**

Let $\{\psi_{j}:j\in J\}$ be a sequence of elements in Hilbert $C^{*}$-module $\mathcal{H}$. Let $W=\{\omega_{j}:j\in J\}$ be a sequence of positive and semi-normalized weights. Then the following properties are equivalent:

1. (1)

$\{\psi_{j}:j\in J\}$* is a frame;* 2. (2)

$M_{w,\Psi}$* is a positive and invertible operator;* 3. (3)

There are constants $C,D>0$ such that for all $f\in\mathcal{H}$

[TABLE]

i.e. the pair $\{\omega_{j}:j\in J\},\{\psi_{j}:j\in J\}$ forms a weighted frame; 4. (4)

$\{\sqrt{\omega_{j}}\psi_{j}:j\in J\}$* is a frame;* 5. (5)

$M_{w,\Psi}$* is a positive and invertible operator for any positive, semi-normalized sequence $W^{\prime}=\{\omega^{{}^{\prime}}_{j}\}_{j\in J}$;* 6. (6)

$(\omega_{j}\psi_{j})$* is a frame, i.e. the pair $\{\omega_{j}:j\in J\},\{\psi_{j}:j\in J\}$ forms a weighted frame.*

Proof.

It is similar to Theorem 4.5 of [5]. ∎

Acknowledgement: The authors would like to thank the refree for useful and helpful comments and suggestions.