An appropriate representation space for controlled G-frames

TL;DR

This paper introduces a new representation space for controlled g-frames, defines associated synthesis and analysis operators, and explores their properties, including duals and inequalities, with implications for frame theory.

Contribution

It proposes an appropriate representation space for controlled g-frames and analyzes the properties of synthesis and analysis operators, including duals and trace class operators.

Findings

The composition of synthesis and analysis operators forms a trace class operator.

The canonical controlled g-dual yields minimal norm expansion coefficients.

Extended known equalities and inequalities for controlled g-frames.

Abstract

In this paper, motivating the range of operators, we propose an appropriate representation space to introduce synthesis and analysis operators of controlled g-frames and discuss the properties of these operators. Especially, we show that the operator obtained by the composition of the synthesis and analysis operators of two controlled g-Bessel sequence is a trace class operator. Also, we define the canonical controlled g-dual and show that this dual gives rise to expand coefficients with the minimal norm. Finally, we extend some known equality and inequalities for controlled g-frames.

Equations

$$\underset{\Vert f\Vert =1}{\inf }⟨vf,f⟩\ge \frac{4ac-b^2}{4a}.$$

$$\underset{\Vert f\Vert =1}{\sup }⟨vf,f⟩\le \frac{4ac-b^2}{4a}.$$

$$u-v=u^2-v^2.$$

$$u{u}^{†}x=x,x\in \mathcal{R}(u).$$

$$A\Vert f{\Vert }^2\le \sum _{i\in \mathbb{I}}\Vert {\Lambda }_{i}f{\Vert }^2\le B\Vert f{\Vert }^2,f\in H.$$

$$T_{\Lambda }:$$

$$T_{\Lambda }(\{f_i$$

$$\displaystyle(\sum_{i\in\mathbb{I}}\oplus H_{i})_{\ell^{2}}=\big{\{}\{f_{i}\}_{i\in\mathbb{I}}\ |\ f_{i}\in H_{i}\ ,\ \sum_{i\in\mathbb{I}}\|f_{i}\|^{2}<\infty\big{\}},$$

$$S_{\Lambda }f=T_{\Lambda }{T}_{\Lambda }^{\ast }f=\sum _{i\in \mathbb{I}}{\Lambda }_i^{\ast }{\Lambda }_{i}f,f\in H,$$

$$A_C\Vert f{\Vert }^2\le \sum _{i\in \mathbb{I}}⟨f,f_i⟩⟨C{f}_i,f⟩\le B_C\Vert f{\Vert }^2.$$

$$A_{C{C}^{\prime }}\Vert f{\Vert }^2\le \sum _{i\in \mathbb{I}}⟨{\Lambda }_i{C}^{\prime }f,{\Lambda }_{i}Cf⟩\le B_{C{C}^{\prime }}\Vert f{\Vert }^2.$$

$$A_{C{C}^{\prime }}\Vert f{\Vert }^2\le \sum _{i\in \mathbb{I}}\Vert (C^{\ast }{\Lambda }_i^{\ast }{\Lambda }_i{C}^{\prime }{)}^{\frac{1}{2}}f{\Vert }^2\le B_{C{C}^{\prime }}\Vert f{\Vert }^2,f\in H.$$

$$K:=\big{\{}\{(C^{*}\Lambda^{*}_{i}\Lambda_{i}C^{\prime})^{\frac{1}{2}}f\}_{i\in\mathbb{I}}:\ f\in H\big{\}}\subset(\sum_{i\in\mathbb{I}}\oplus H_{i})_{\ell^{2}}.$$

$$T_{C{C}^{\prime }}:K⟶H,$$

$$T_{C{C}^{\prime }}(\{(C^{\ast }{\Lambda }_i^{\ast }{\Lambda }_i{C}^{\prime }{)}^{\frac{1}{2}}f{\}}_{i\in \mathbb{I}})$$

$$T_{C{C}^{\prime }}^{\ast }$$

$$T_{C{C}^{\prime }}^{\ast }(f)=$$

$$S_{C{C}^{\prime }}f=T_{C{C}^{\prime }}{T}_{C{C}^{\prime }}^{\ast }f=\sum _{i\in \mathbb{I}}{C}^{\ast }{\Lambda }_i^{\ast }{\Lambda }_i{C}^{\prime }f,f\in H.$$

$$⟨S_{C{C}^{\prime }}f,f⟩=\sum _{i\in \mathbb{I}}⟨{\Lambda }_i{C}^{\prime }f,{\Lambda }_{i}Cf⟩,f\in H,$$

$$A_{C{C}^{\prime }}I{d}_H\le S_{C{C}^{\prime }}\le B_{C{C}^{\prime }}I{d}_H.$$

$$f=\sum _{i\in \mathbb{I}}{C}^{\ast }{\Lambda }_i^{\ast }{\Lambda }_i{C}^{\prime }{S}_{C{C}^{\prime }}^{-1}f=\sum _{i\in \mathbb{I}}{S}_{C{C}^{\prime }}^{-1}{C}^{\ast }{\Lambda }_i^{\ast }{\Lambda }_i{C}^{\prime }f,f\in H.$$

$$f=S_{C{C}^{\prime }}^{-\frac{1}{2}}{S}_{C{C}^{\prime }}{S}_{C{C}^{\prime }}^{-\frac{1}{2}}f=\sum _{i\in \mathbb{I}}{S}_{C{C}^{\prime }}^{-\frac{1}{2}}{C}^{\ast }{\Lambda }_i^{\ast }{\Lambda }_i{C}^{\prime }{S}_{C{C}^{\prime }}^{-\frac{1}{2}}f.$$

$$\Vert f{\Vert }^2=⟨f,f⟩=\sum _{i\in \mathbb{I}}⟨{\Lambda }_i{S}_{C{C}^{\prime }}^{-\frac{1}{2}}{C}^{\prime }f,{\Lambda }_i{S}_{C{C}^{\prime }}^{-\frac{1}{2}}Cf⟩.$$

$$T_{C{C}^{\prime }}:K⟶H,$$

$$T_{C{C}^{\prime }}(\{(C^{\ast }{\Lambda }_i^{\ast }{\Lambda }_i{C}^{\prime }{)}^{\frac{1}{2}}f{\}}_{i\in \mathbb{I}})$$

$$\sum _{i\in \mathbb{I}}⟨{\Lambda }_i{C}^{\prime }f,{\Lambda }_{i}Cf⟩$$

$$\displaystyle=\big{\langle}T_{CC^{\prime}}(\{(C^{*}\Lambda^{*}_{i}\Lambda_{i}C^{\prime})^{\frac{1}{2}}f\}_{i\in\mathbb{I}}),f\big{\rangle}$$

$$\le \Vert T_{C{C}^{\prime }}\Vert \Vert \{(C^{\ast }{\Lambda }_i^{\ast }{\Lambda }_i{C}^{\prime }{)}^{\frac{1}{2}}f{\}}_{i\in \mathbb{I}}\Vert \Vert f\Vert .$$

$$\Vert \{(C^{\ast }{\Lambda }_i^{\ast }{\Lambda }_i{C}^{\prime }{)}^{\frac{1}{2}}f{\}}_{i\in \mathbb{I}}{\Vert }^2=\sum _{i\in \mathbb{I}}⟨{\Lambda }_i{C}^{\prime }f,{\Lambda }_{i}Cf⟩.$$

$$\sum _{i\in \mathbb{I}}⟨{\Lambda }_i{C}^{\prime }f,{\Lambda }_{i}Cf⟩\le B_{C{C}^{\prime }}\Vert f{\Vert }^2,$$

$$T_{C{C}^{\prime }}:(\{(C^{\ast }{\Lambda }_i^{\ast }{\Lambda }_i{C}^{\prime }{)}^{\frac{1}{2}}f{\}}_{i\in \mathbb{I}})⟼\sum _{i\in \mathbb{I}}{C}^{\ast }{\Lambda }_i^{\ast }{\Lambda }_i{C}^{\prime }f$$

$$\Vert f{\Vert }^4$$

$$=|⟨T_{C{C}^{\prime }}{T}_{C{C}^{\prime }}^{†}f,f⟩{|}^2$$

$$=|⟨T_{C{C}^{\prime }}^{†}f,T_{C{C}^{\prime }}^{\ast }f⟩{|}^2$$

Full text

1. Introduction and Preliminaries

Frames were first introduced in the context of non-harmonic Fourier series by Duffin and Schaeffer [5]. During the last $20$s the theory of frames has been developed rapidly and because of the abundant use of frames in engineering and applied sciences, many generalization of frames have come into play.

G-frames that include the concept of ordinary frames have been introduced by Sun [15] and improved by many authors [6, 8, 10, 14]. Controlled frames have been improved recently to improve the numerical efficiency of interactive algorithms that inverts the frame operator [2]. Following that, controlled frames have been generalized to another kinds of frames [6, 7, 9, 13, 14, 12, 8].

In this paper, motivating the concept of g-frames and controlled frames we define controlled g-frames. In Section 2, imagined the range of an operator, a new representation space is introduced such that the synthesis and analysis operators could be defined. In Section 3, controlled g- dual frames and canonical controlled g-dual frames are introduced and shown that canonical g-dual gives rise to expand coefficients with the minimal norm. Finally, some equalities and inequalities are presented for controlled g-frames and especially for their operators in Section 4.

Throughout this paper, $H$ is a separable Hilbert space, $\{H_{i}\}_{i\in\mathbb{I}}$ is the collection of Hilbert spaces, $\mathcal{B}(H,K)$ is the family of all linear bounded operators from $H$ into $K$ and $\mathcal{GL}(H)$ is the set of all bounded linear operators which have bounded inverses.

At first, we collect some definitions and basic results that are needed in the paper.

Lemma 1.1**.**

([11]) Let $u\in\mathcal{B}(H)$ be a self-adjoint operator and $v:=au^{2}+bu+c$ where $a,b,c\in\mathbb{R}$.

1. (i)

If $a>0$, then

[TABLE] 2. (ii)

If $a<0$, then

[TABLE]

Lemma 1.2**.**

([1]) If $u,v$ are operators on $H$ satisfying $u+v=id_{H}$, then

[TABLE]

If an operator $u$ has closed range, then there exists a right-inverse operator $u^{\dagger}$ (pseudo-inverse of $u$) in the following senses (see [4]).

Lemma 1.3**.**

Let $u\in\mathcal{B}(K,H)$ be a bounded operator with closed range $\mathcal{R}_{u}$. Then there exists a bounded operator $u^{\dagger}\in\mathcal{B}(H,K)$ for which

[TABLE]

Definition 1.4**.**

(g-frame) A family $\Lambda:=\{\Lambda_{i}\in\mathcal{B}(H,H_{i})\}_{i\in\mathbb{I}}$ is called a g-frame for $H$ with respect to $\{H_{i}\}_{i\in\mathbb{I}}$, if there exist $0<A\leq B<\infty$ such that

[TABLE]

If only the second inequality in (1.1) satisfy, then we say that $\{\Lambda_{i}\}_{i\in\mathbb{I}}$ is a g-Bessel sequence with upper bound $B$.

If $\Lambda$ is a g-Bessel sequence, then the synthesis and analysis operators are defined by

[TABLE]

where

[TABLE]

and, the g-frame operator is given by

[TABLE]

which is positive, self-adjoint and invertible (see [15]).

2. Controlled g-frames and their operators

Controlled frames for spherical wavelets were introduced in [3] to get a numerically more efficient approximation algorithm. In this section by extending the concept of controlled frames and g-frames, we define the concept of controlled g-frames and construct an appropriate representation space to organize the synthesis and analysis operators.

Definition 2.1**.**

[2] Let $C\in\mathcal{GL}(H)$. We say that $F:=\{f_{i}\}_{i\in\mathbb{I}}$ is a $C$-controlled frame for $H$ if there exist $0<A_{C}\leq B_{C}<\infty$ such that for each $f\in H$

[TABLE]

Definition 2.2**.**

Let $C,C^{\prime}\in\mathcal{GL}(H)$. We say that $\Lambda:=\{\Lambda_{i}\in\mathcal{B}(H,H_{i})\}_{i\in\mathbb{I}}$ is a $(C,C^{\prime})$-controlled g-frame for $H$ if there exist $0<A_{CC^{\prime}}\leq B_{CC^{\prime}}<\infty$ such that for each $f\in H$

[TABLE]

For simplicity, we use a notation $CC^{\prime}$ instead of $(C,C^{\prime})$. We call $\Lambda$ a Parseval $CC^{\prime}$-controlled g-frame if $A_{CC^{\prime}}=B_{CC^{\prime}}=1$. When the right hand inequality of (2.2) holds, then $\Lambda$ is called a $CC^{\prime}$-controlled g-Bessel sequence for $H$ with bound $B_{C}$.

If $\Lambda$ is a $CC^{\prime}$-controlled g-frame for $H$ and $C^{*}\Lambda^{*}_{i}\Lambda_{i}C^{\prime}$ is positive for each $i\in\mathbb{I}$, then we have

[TABLE]

Consider a proper representation space by

[TABLE]

Since $K$ is a closed subspace of $(\sum_{i\in\mathbb{I}}\oplus H_{i})_{\ell^{2}}$, we can define the synthesis and analysis operators of $CC^{\prime}$-controlled g-frames by

[TABLE]

and

[TABLE]

Thus, the $CC^{\prime}$-controlled g-frame operator is given by

[TABLE]

So,

[TABLE]

and

[TABLE]

Therefore, $S_{CC^{\prime}}$ is a positive, self-adjoint and invertible operator (see [14] and [9]). Thus, since $f=S_{CC^{\prime}}S^{-1}_{CC^{\prime}}f=S^{-1}_{CC^{\prime}}S_{CC^{\prime}}f$, we have

[TABLE]

Remark 2.3*.*

We introduce a Parseval $CC^{\prime}$-controlled g-frame for $H$ by the $CC^{\prime}$-controlled g-frame operator. Suppose that $\Lambda$ is a $CC^{\prime}$-controlled g-frame for $H$. Since $S_{CC^{\prime}}$(or $S_{CC^{\prime}}^{-1}$) is positive in $\mathcal{B}(H)$ and $\mathcal{B}(H)$ is a $C^{*}$-algebra, then there exists a unique positive square root $S^{\frac{1}{2}}_{CC^{\prime}}$ (or $S^{-\frac{1}{2}}_{CC^{\prime}}$) which commutes with $S_{CC^{\prime}}$ and $S_{CC^{\prime}}^{-1}$. Therefore, for any $f\in H$ we can write

[TABLE]

Now, assume that $S^{-\frac{1}{2}}_{CC^{\prime}}$ commutes with $C,C^{\prime}$. Then we get

[TABLE]

Hence, $\{\Lambda_{i}S^{-\frac{1}{2}}_{CC^{\prime}}\}_{i\in\mathbb{I}}$ is a Parseval $CC^{\prime}$-controlled g-frame for $H$.

Theorem 2.4**.**

A sequence $\Lambda$ is a $CC^{\prime}$-controlled g-Bessel sequence for $H$ with bound $B_{CC^{\prime}}$ if and only if the operator

[TABLE]

is a well-defined and bounded operator with $\|T_{CC^{\prime}}\|\leq\sqrt{B_{CC^{\prime}}}$.

Proof.

We only need to prove the sufficient condition. Let $T_{CC^{\prime}}$ be a well-defined and bounded operator with $\|T_{CC^{\prime}}\|\leq\sqrt{B_{CC^{\prime}}}$. For each $f\in H$ we have

[TABLE]

But

[TABLE]

It follows that

[TABLE]

and this means that $\Lambda$ is a $CC^{\prime}$-controlled g-Bessel sequence. ∎

Theorem 2.5**.**

A sequence $\Lambda$ is a $CC^{\prime}$-controlled g-frame for $H$ if and only if

[TABLE]

is a well-defined and surjective operator.

Proof.

First, suppose that $\Lambda$ is a $CC^{\prime}$-controlled g-frame for $H$. Since, $S_{CC^{\prime}}$ is a surjective operator, so $T_{CC^{\prime}}$. For the opposite implication, by Theorem 2.4, $T_{CC^{\prime}}$ is a well-defined and bounded operator. So, $\Lambda$ is a $CC^{\prime}$-controlled g-Bessel sequence. Now, for each $f\in H$, we have $f=T_{CC^{\prime}}T^{\dagger}_{CC^{\prime}}f$. Hence,

[TABLE]

We conclude that

[TABLE]

∎

Theorem 2.6**.**

Let $\Lambda=\{\Lambda_{i}\in\mathcal{B}(H,H_{i})\}_{i\in\mathbb{I}}$ and $\Theta:=\{\Theta_{i}\in\mathcal{B}(H,H_{i})\}_{i\in\mathbb{I}}$ be two $CC^{\prime}$-controlled g-Bessel sequence for $H$ with bounds $B_{1}$ and $B_{2}$, respectively. If $T_{\Lambda}$ and $T_{\Theta}$ are their synthesis operators such that $T_{\Lambda}T^{*}_{\Theta}=Id_{H}$, then both $\Lambda$ and $\Theta$ are $CC^{\prime}$-controlled g-frames for $H$.

Proof.

For each $f\in H$, we have

[TABLE]

Hence,

[TABLE]

and $\Lambda$ is a $CC^{\prime}$-controlled g-frame. Similarly, $\Theta$ is a $CC^{\prime}$-controlled g-frame with lower bound $B^{-1}_{1}$. ∎

Theorem 2.7**.**

Let $\Lambda=\{\Lambda_{i}\in\mathcal{B}(H,H_{i})\}_{i\in\mathbb{I}}$ and $\Theta:=\{\Theta_{i}\in\mathcal{B}(H,H_{i})\}_{i\in\mathbb{I}}$ be two $CC^{\prime}$-controlled g-Bessel sequence for $H$ with bounds $B_{1}$ and $B_{2}$, respectively where $|\mathbb{I}|<\infty$. If $\Phi:=T_{\Lambda}T^{*}_{\Theta}$, then $\Phi$ is a trace class operator.

Proof.

Suppose that $\Phi=u|\Phi|$ is the polar decomposition of $\Phi$ where $u\in\mathcal{B}(H)$ is a partial isometry. So, $|\Phi|=u^{*}\Phi$. Let $\{e_{i}\}_{i\in\mathbb{I}}$ be an orthonormal basis for $H$. We have

[TABLE]

∎

3. Controlled g-dual frames

In this section by considering that $C=C^{\prime}$ and $\Lambda$ is a $C^{2}$-controlled g-frame for $H$, we define a canonical controlled g-dual and show that this canonical dual is a g-frame and gives rise to expand coefficients with the minimal norm.

Definition 3.1**.**

Suppose that $\Lambda=\{\Lambda_{i}\in\mathcal{B}(H,H_{i})\}_{i\in\mathbb{I}}$ and $\widetilde{\Lambda}:=\{\widetilde{\Lambda}_{i}\in\mathcal{B}(H,H_{i})\}_{i\in\mathbb{I}}$ are two $CC^{\prime}$-controlled g-Bessel sequence for $H$ with synthesis operators $T_{\Lambda}$ and $T_{\widetilde{\Lambda}}$, respectively. We say that $\widetilde{\Lambda}$ is a $CC^{\prime}$-controlled g-dual of $\Lambda$ if

[TABLE]

In this case $\Lambda,\widetilde{\Lambda}$ are said $CC^{\prime}$-controlled g-dual pair also.

The proof of the following is straightforward.

Proposition 3.2**.**

If $\Lambda,\widetilde{\Lambda}$ are $CC^{\prime}$-controlled g-dual pair, then the following statements are equivalent:

1. (i)

$T_{\Lambda}T^{*}_{\widetilde{\Lambda}}=Id_{H}$; 2. (ii)

$T_{\widetilde{\Lambda}}T^{*}_{\Lambda}=Id_{H}$; 3. (iii)

$\langle f,g\rangle=\langle T^{*}_{\widetilde{\Lambda}}f,T^{*}_{\Lambda}g\rangle$, $f,g\in H$.

Also, for every $f\in H$, we have

[TABLE]

Now, we want to present the canonical controlled g-dual by (2.3) in the case that $C=C^{\prime}$ and $\Lambda$ is a $C^{2}$-controlled g-frame for $H$. Let $\Gamma_{i}:=\Lambda_{i}CS_{C}^{-1}$. Therefore, for each $f\in H$

[TABLE]

We show that $\Gamma:=\{\Gamma_{i}\}_{i\in\mathbb{I}}$ is a g-frame for $H$. Let $f\in H$ and $A_{C},B_{C}$ be the frame bounds of $\Lambda$. Then

[TABLE]

On the other hand,

[TABLE]

Finally, we conclude that

[TABLE]

The following theorem shows that the canonical controlled g-dual gives rise to expand coefficients with the minimal norm.

Theorem 3.3**.**

Let $\Lambda=\{\Lambda_{i}\in\mathcal{B}(H,H_{i})\}_{i\in\mathbb{I}}$ be a $C^{2}$-controlled g-frame for $H$ and $\Gamma_{i}=\Lambda_{i}CS_{C}^{-1}$. If $f$ has a representation $f=\sum_{i\in\mathbb{I}}C^{*}\Lambda_{i}^{*}g_{i}$, for some $g_{i}\in H_{i}$. Then we have

[TABLE]

Proof.

Assume that $f\in H$. We get by (3.2)

[TABLE]

Therefore, \mbox{Im}\Big{(}\sum_{i\in\mathbb{I}}\langle g_{i},\Gamma_{i}f\rangle\Big{)}=0 and the conclusion follows. ∎

4. Some equalities and inequalities

In this section, we extend some known equalities and inequalities for controlled g-frames. Assume that $\Lambda,\widetilde{\Lambda}$ are $CC^{\prime}$-controlled g-dual pair and $\mathbb{J}\subset\mathbb{I}$. We define

[TABLE]

It is clear that $S_{\mathbb{J}}\in\mathcal{B}(H)$ and $S_{\mathbb{J}}+S_{\mathbb{J}^{c}}=Id_{H}$ where $\mathbb{J}^{c}$ is the complement of $\mathbb{J}$. Indeed, if $B_{1}$ and $B_{2}$ are the bounds of $\Lambda$ and $\widetilde{\Lambda}$ respectively, then

[TABLE]

So, $S_{\mathbb{J}}$ is bounded.

Theorem 4.1**.**

If $f\in H$ then,

[TABLE]

Proof.

Let $f\in H$. We obtain

[TABLE]

Now, the proof is completed. ∎

Corollary 4.2**.**

*If $\Lambda$ is a $CC^{\prime}$-controlled Parseval g-frame for $H$, then *

[TABLE]

Moreover,

[TABLE]

Proof.

If $f\in H$, we obtain

[TABLE]

Now, by Lemma 1.1 for $a=1$, $b=-1$ and $c=1$ the inequality holds. ∎

Corollary 4.3**.**

If $\Lambda$ is a $CC^{\prime}$-controlled Parseval g-frame for $H$, then

[TABLE]

Proof.

We have $S_{\mathbb{J}}S_{\mathbb{J}^{c}}=S_{\mathbb{J}^{c}}S_{\mathbb{J}}$. Then $0\leq S_{\mathbb{J}}S_{\mathbb{J}^{c}}=S_{\mathbb{J}}-S_{\mathbb{J}}^{2}$. Also, by Lemma 1.1, we get

[TABLE]

∎

Theorem 4.4**.**

*Let $\Lambda$ be a $CC^{\prime}$-controlled g-frame with $CC^{\prime}$-controlled g-frame operator $S_{CC^{\prime}}$. Suppose that $S^{-\frac{1}{2}}_{CC^{\prime}}$ commutes with $C,C^{\prime}$. Then for each $f\in H$, *

[TABLE]

Proof.

Via Remark 2.3 and Corollary 4.2, if $\Theta_{i}:=\Lambda_{i}S^{-\frac{1}{2}}_{CC^{\prime}}$, then

[TABLE]

and also

[TABLE]

Now, by replacing $S_{CC^{\prime}}^{\frac{1}{2}}f$ instead of $f$ in above formulas, the proof is evident. ∎

Corollary 4.5**.**

Let $\Lambda$ be a $CC^{\prime}$-controlled g-frame with $CC^{\prime}$-controlled g-frame operator $S_{CC^{\prime}}$. Suppose that $S^{-\frac{1}{2}}_{CC^{\prime}}$ commutes with $C,C^{\prime}$. Then

[TABLE]

Proof.

In the proof of Theorem 4.4, we showed that

[TABLE]

By Corollary 4.3 we get

[TABLE]

So,

[TABLE]

and it completes the proof. ∎

Corollary 4.6**.**

Let $\Lambda$ be a $CC^{\prime}$-controlled g-frame with $CC^{\prime}$-controlled g-frame operator $S_{CC^{\prime}}$ . Suppose that $S^{-\frac{1}{2}}_{CC^{\prime}}$ commutes $C,C^{\prime}$. Then for each $f\in H$,

[TABLE]

Proof.

Applying Theorem 4.4 and Corollary 4.2, we obtain

[TABLE]

∎

Theorem 4.7**.**

Let $\Lambda$ be a Parseval $CC^{\prime}$-controlled g-frame for $H$. Then

1. (i)

$0\leq S_{\mathbb{J}}-S_{\mathbb{J}}^{2}\leq\dfrac{1}{4}id_{H}$. 2. (ii)

$\dfrac{1}{2}id_{H}\leq S_{\mathbb{J}}^{2}+S_{\mathbb{J}^{c}}^{2}\leq\dfrac{3}{2}id_{H}$.

Proof.

(i). Since $S_{\mathbb{J}}+S_{\mathbb{J}^{c}}=id_{H}$, then $S_{\mathbb{J}}S_{\mathbb{J}^{c}}+S^{2}_{\mathbb{J}^{c}}=S_{\mathbb{J}^{c}}$. Thus

[TABLE]

But, $\Lambda$ is a Parseval $CC^{\prime}$-controlled g-frame, so $0\leq S_{\mathbb{J}}S_{\mathbb{J}^{c}}=S_{\mathbb{J}}-S_{\mathbb{J}}^{2}$. On the other hand, by Lemma 1.2, we get

[TABLE]

(ii). Since $S_{\mathbb{J}}S_{\mathbb{J}^{c}}=S_{\mathbb{J}^{c}}S_{\mathbb{J}}$, so by Lemma 1.2

[TABLE]

and we get the right inequality by Lemma 1.2 and $0\leq S_{\mathbb{J}}-S_{\mathbb{J}}^{2}$,

[TABLE]

∎