Controlled g-frames and dual g-frames in Hilbert spaces

Abstract

As generalizations of g-frames and controlled frames, the theory of controlled g-frames has been deeply studied. This paper addresses the controlled g-frames and dual g-frames in Hilbert spaces. We first present some equivalent characterizations of controlled g-frames. Then, we introduce the concepts of controlled dual g-frames and controlled dual g-frames operator, get some properties of them. Finally, we obtain some characterizations of the controlled dual g-frames for a given controlled g-frame by the method of operator theory.

1 Introduction

A sequence $\{f_{j}\}_{j\in J}$ in a separable Hilbert space $\mathcal{H}$ is called a frame if there exist $0< A\leq B<\infty$ such that

$$ A \Vert f \Vert ^{2}\leq \sum_{j\in J} \bigl\vert \langle f, f_{j}\rangle \bigr\vert ^{2} \leq B \Vert f \Vert ^{2} $$

for all $f\in \mathcal{H}$. The concept of frames was introduced by Gabor in 1946 and Duffin and Schaeffer in 1952. Gabor in [12] proposed the idea of decomposing a general signal in terms of elementary signals, and Duffin and Schaeffer in [10] abstracted “these elementary signals” as the notion of frame. The frame theory has been developing rapidly since Daubechies, Grossmann, and Meyer [9] had put forward the definition of frames for Hilbert spaces formally in 1986. So far, the theory of frame has achieved fruitful success in pure mathematics, science, and engineering [4, 5, 8, 13, 14, 21, 24]. In the last decades, various generalizations of frame have been put forward for special purposes such as frame of subspaces [6], fusion frame [7], bounded quasi-projector [11], and g-frame [22]. In particular, among these generalizations, a g-frame covers all others, and the research of g-frames has obtained many results [16, 23, 25]. Controlled frames have been introduced to improve the numerical efficiency of iterative algorithms for inverting the frame operator on abstract Hilbert spaces [2]. A sequence $\{f_{j}\}_{j\in J}\subset \mathcal{H}$ is called a C-controlled frame if there exist positive constants $0< A_{2}\leq B_{2}<\infty$ such that

$$ A_{2} \Vert f \Vert ^{2}\leq \sum _{j\in J}\langle f, f_{j}\rangle \langle Cf_{j}, f\rangle \leq B_{2} \Vert f \Vert ^{2} $$

for all $f\in \mathcal{H}$, where $C\in \mathcal{GL}(\mathcal{H})$. However, they are only used as a tool to study spherical wavelets [3]. Later, some scholars noticed that these frames can give a generalized way to check the frame conditions while offering numerical advantages in the sense of preconditioning. Since then, controlled frames have been widely studied [15, 17–20]. Rahimi et al. in [18] first introduced the notion of controlled g-frames (see Definition 2.3), which is an extension of g-frames and controlled frames.

Inspired by the above research, in this paper we address the characterization of controlled g-frames and controlled dual g-frames, and it is organized as follows: In Sect. 2, we recall some basic notions, properties, and related results. Section 3 is devoted to the characterization of controlled g-frames, we obtain some equivalent conditions of controlled g-frames. In Sect. 4, we introduce the notion of controlled dual frames in Hilbert spaces and obtain some characterizations of the controlled dual g-frames for a given controlled g-frame by the method of operator theory.

2 Preliminaries

We begin this section with some basic notions and results of g-frames (see [8, 18, 20, 22, 25] for details).

Given separable Hilbert spaces $\mathcal{H}$ and $\mathcal{V}$, let $\{\mathcal{V}_{j}:j\in J\}$ be a sequence of closed subspaces of $\mathcal{V}$ with J being a subset of integers $\mathbb{Z}$. The identity operator on $\mathcal{H}$ is denoted by $I_{\mathcal{H}}$. The set of all bounded linear operators from $\mathcal{H}$ into $\mathcal{V}_{j}$ is denoted by $L(\mathcal{H}, \mathcal{V}_{j})$. As a special case, $L(\mathcal{H})$ is a collection of all bounded linear operators on $\mathcal{H}$. The set of all bounded linear operators on $\mathcal{H}$ with a bounded inverse is denoted by $\mathcal{GL}(\mathcal{H})$. If $P, Q\in \mathcal{GL}(\mathcal{H})$, then $P^{\ast}$, $P^{-1}$, and PQ are also in $\mathcal{GL}(\mathcal{H})$. Let $\mathcal{GL}^{+}(\mathcal{H})$ be the set of all positive operators in $\mathcal{GL}(\mathcal{H})$. A bounded operator $P: \mathcal{H}\rightarrow \mathcal{H}$ is positive if $\langle Pf, f\rangle >0$ for all $f\neq 0$. In a complex Hilbert space, every bounded positive operator is self-adjoint. In addition, as a technical condition, we also assume that any two positive operators involved in this paper commutate with each other. Define

$$ \bigoplus_{j\in J}\mathcal{V}_{j}= \biggl\{ \{a_{j}\}_{j\in J}:a_{j} \in \mathcal{V}_{j}, \bigl\Vert \{a_{j}\}_{j\in J} \bigr\Vert ^{2} = \sum_{j\in J} \Vert a_{j} \Vert ^{2}< \infty \biggr\} . $$

Then $\bigoplus_{j\in J}\mathcal{V}_{j}$ is a Hilbert space under the following inner product:

$$ \bigl\langle \{a_{j}\}_{j\in J}, \{b_{j} \}_{j\in J} \bigr\rangle = \sum_{j\in J} \langle a_{j}, b_{j} \rangle \quad \text{for } \{a_{j} \}_{j\in J}, \{b_{j}\}_{j\in J}\in \bigoplus _{j\in J} \mathcal{V}_{j}. $$

Suppose that $\{e_{j,k}\}_{k\in K_{j}}$ is an orthonormal basis (simply o. n. b.) for $\mathcal{V}_{j}$, where $K_{j}\subset \mathbb{Z}$, $j\in J$. Define $\tilde{e}_{j,k}=e_{j,k}\delta _{j}$, where δ is the Kronecker symbol. Then $\{\tilde{e}_{j,k}\}_{j\in J,k\in K_{j}}$ is an o. n. b. for $\bigoplus_{j\in J}\mathcal{V}_{j}$ (see [25]).

Definition 2.1

([22])

A sequence $\{\Lambda _{j}\in L(\mathcal{H}, \mathcal{V}_{j})\}_{j\in J}$ is called a g-frame for $\mathcal{H}$ with respect to (simply w. r. t.) $\{\mathcal{V}_{j}\}_{j\in J}$ if

$$ A \Vert f \Vert ^{2}\leq \sum _{j\in J} \Vert \Lambda _{j}f \Vert ^{2} \leq B \Vert f \Vert ^{2} $$

(2.1)

for all $f\in \mathcal{H}$ and some positive constants $A\leq B$. The numbers A, B are called the frame bounds. If only the right-hand inequality of (2.1) is satisfied, $\{\Lambda _{j}\}_{j\in J}$ is called a g-Bessel sequence for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$ with bound B. If $A=B=\lambda$, $\{\Lambda _{j}\}_{j\in J}$ is called a λ-tight g-frame. In addition, if $\lambda =1$, $\{\Lambda _{j} \}_{j\in J}$ is called a Parseval g-frame.

Definition 2.2

([25])

Let $\{\Lambda _{j}\}_{j\in J}$ be a g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$. A g-frame $\{\Gamma _{j}\}_{j\in J}$ is called an alternate dual g-frame for $\{\Lambda _{j}\}_{j\in J}$ if

$$ f=\sum_{j\in J}\Gamma _{j}^{\ast} \Lambda _{j}f\quad \text{for } f \in \mathcal{H}. $$

Moreover, $\{\Lambda _{j}\}_{j\in J}$ is also an alternate dual g-frame for $\{\Gamma _{j}\}_{j\in J}$, that is,

$$ f=\sum_{j\in J}\Lambda _{j}^{\ast} \Gamma _{j}f\quad \text{for } f \in \mathcal{H}. $$

Definition 2.3

([8])

Let $P, Q\in \mathcal{GL}^{+}(\mathcal{H})$. A sequence $\{\Lambda _{j}\}_{j\in J}$ is called a $(P, Q)$-controlled g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$. If there exist two positive constants A and B such that

$$ A \Vert f \Vert ^{2}\leq \sum _{j\in J}\langle \Lambda _{j}Pf, \Lambda _{j}Qf \rangle \leq B \Vert f \Vert ^{2},\quad \forall f \in \mathcal{H}. $$

(2.2)

We call A and B the lower and upper frame bounds for $(P, Q)$-controlled g-frame, respectively.

If the right-hand side of (2.2) holds, then $\{\Lambda _{j}\}_{j\in J}$ is called a $(P, Q)$-controlled g-Bessel sequence for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$.

If $Q=I_{\mathcal{H}}$, then we call $\{\Lambda _{j}\}_{j\in J}$ a P-controlled g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$.

If $P=Q$, then we call $\{\Lambda _{j}\}_{j\in J}$ a $P^{2}(\text{or }(P, P))$-controlled g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$.

Lemma 2.1

([8])

Every bounded and positive operator $P: \mathcal{H}\rightarrow \mathcal{H}$ has a unique bounded and positive square root W. If P is self-adjoint, then W is self-adjoint. If P is invertible, then W is also invertible.

For a $(P, Q)$-controlled g-Bessel sequence $\{\Lambda _{j}\}_{j\in J}$ with bound B, the operator $T_{P\Lambda Q}$

$$ T_{P\Lambda Q}: \bigoplus_{j\in J} \mathcal{V}_{j} \rightarrow \mathcal{H}, \qquad T_{P\Lambda Q}F=\sum _{j\in J}(PQ)^{ \frac{1}{2}}\Lambda _{j}^{\ast}f_{j}, \quad \forall F=\{f_{j}\}_{j\in J} \in \bigoplus _{j\in J}\mathcal{V}_{j} $$

is well defined, and its adjoint is given by

$$ T_{P\Lambda Q}^{\ast}: \mathcal{H} \rightarrow \bigoplus _{j \in J}\mathcal{V}_{j}, \qquad T_{P\Lambda Q}^{\ast}f= \bigl\{ \Lambda _{j}(QP)^{ \frac{1}{2}}f\bigr\} _{j\in J},\quad \forall f\in \mathcal{H}. $$

$T_{P\Lambda Q}$ is called the synthesis operator and $T_{P\Lambda Q}^{\ast}$ is called the analysis operator of $\{\Lambda _{j}\}_{j\in J}$. For a $(P, Q)$-controlled g-frame $\{\Lambda _{j}\}_{j\in J}$ with bounds A and B, the operator

$$ S_{P\Lambda Q}: \mathcal{H}\rightarrow \mathcal{H},\qquad S_{P\Lambda Q}f= \sum_{j\in J}Q\Lambda _{j}^{\ast} \Lambda _{j}Pf,\quad \forall f \in \mathcal{H} $$

is called the frame operator of $\{\Lambda _{j}\}_{j\in J}$. From the definition, $S_{P\Lambda Q}=PS_{\Lambda}Q$ is positive and invertible, where $S_{\Lambda}$ is a frame operator of g-frame $\{\Lambda _{j}\}_{j\in J}$, and it is bounded, invertible, self-adjoint, positive, and $AI_{H} \leq S_{\Lambda}\leq BI_{H}$. Let $\tilde{\Lambda}_{j}=\Lambda _{j}S_{\Lambda}^{-1}$, then $\{\tilde{\Lambda}_{j}\}_{j\in J}$ is a g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$ with frame operator $S_{\Lambda}^{-1}$ and frame bounds $\frac{1}{B}$ and $\frac{1}{A}$, respectively. $\{\tilde{\Lambda}_{j}\}_{j\in J}$ is called the canonical dual g-frame of $\{\Lambda _{j}\}_{j\in J}$ (see [22]).

Definition 2.4

([20])

Let $\mathcal{H}$ be a Hilbert space and $C\in \mathcal{GL}(\mathcal{H})$. Suppose that $\{\psi _{j}\}_{j\in J}\subseteq \mathcal{H}$ is a C-controlled frame and $\{\phi _{j}\}_{j\in J}\subseteq \mathcal{H}$ is a Bessel sequence. Then $\{\phi _{j}\}_{j\in J}\subseteq \mathcal{H}$ is said to be a C-controlled dual of $\{\psi _{j}\}_{j\in J}\subseteq \mathcal{H}$ if the following condition is satisfied:

$$ f=\sum_{j\in J}\langle f, \phi _{j}\rangle C \psi _{j} $$

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

3 Controlled g-frames in Hilbert spaces

In this section, we present the characterization of controlled dual g-frames, and some equivalent conditions of $(P, Q)$-controlled g-frames are obtained. For this purpose, we first give some equivalent conditions of bounded and positive operators.

Lemma 3.1

([8])

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

• (i) There exist two constants $0< c\leq C<\infty$ such that $cI_{\mathcal{H}}\leq T\leq CI_{\mathcal{H}}$.

• (ii) T is positive and there exist two constants $0< c\leq C<\infty$ such that

  $$ c \Vert f \Vert ^{2}\leq \bigl\Vert T^{\frac{1}{2}}f \bigr\Vert ^{2}\leq C \Vert f \Vert ^{2}. $$

• (iii) $T\in \mathcal{GL}^{+}(\mathcal{H})$.

The following lemma gives a characterization of $(P, Q)$-controlled g-frames in Hilbert space. By Proposition 2.1 in [1], if $P, Q\in \mathcal{GL}^{+}(\mathcal{H})$ and $PQ=QP$, then we have $PQ\in \mathcal{GL}^{+}(\mathcal{H})$.

Lemma 3.2

Let $P, Q\in \mathcal{GL}^{+}(\mathcal{H})$. Then $\{\Lambda _{j}\}_{j\in J}$ is a $(P, Q)$-controlled g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$ if and only if $\{\Lambda _{j}\}_{j\in J}$ is a g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$.

Proof

Suppose that $\{\Lambda _{j}\}_{j\in J}$ is a $(P, Q)$-controlled g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$ with bounds A, B. For any $f\in \mathcal{H}$, we have

$$\begin{aligned} A \Vert f \Vert ^{2}&=A \bigl\Vert (PQ)^{\frac{1}{2}}(PQ)^{-\frac{1}{2}}f \bigr\Vert ^{2} \\ &\leq A \bigl\Vert (PQ)^{\frac{1}{2}} \bigr\Vert ^{2} \bigl\Vert (PQ)^{- \frac{1}{2}}f \bigr\Vert ^{2} \\ &\leq \bigl\Vert (PQ)^{\frac{1}{2}} \bigr\Vert ^{2}\sum _{j\in J} \bigl\langle \Lambda _{j}P(PQ)^{-\frac{1}{2}} f, \Lambda _{j}Q(PQ)^{- \frac{1}{2}}f \bigr\rangle \\ &= \bigl\Vert (PQ)^{\frac{1}{2}} \bigr\Vert ^{2} \bigl\langle QS_{\Lambda}P(PQ)^{- \frac{1}{2}}f, (PQ)^{-\frac{1}{2}}f \bigr\rangle \\ &= \bigl\Vert (PQ)^{\frac{1}{2}} \bigr\Vert ^{2} \bigl\langle S_{\Lambda}P(PQ)^{- \frac{1}{2}}f, Q(PQ)^{-\frac{1}{2}}f \bigr\rangle \\ &= \bigl\Vert (PQ)^{\frac{1}{2}} \bigr\Vert ^{2} \bigl\langle S_{\Lambda}P^{ \frac{1}{2}}(Q)^{-\frac{1}{2}}f, Q^{\frac{1}{2}}(P)^{-\frac{1}{2}}f \bigr\rangle \\ &= \bigl\Vert (PQ)^{\frac{1}{2}} \bigr\Vert ^{2} \bigl\langle (P)^{- \frac{1}{2}}Q^{\frac{1}{2}}S_{\Lambda}P^{\frac{1}{2}}(Q)^{- \frac{1}{2}}f, f \bigr\rangle = \bigl\Vert (PQ)^{\frac{1}{2}} \bigr\Vert ^{2} \langle S_{\Lambda}f, f \rangle . \end{aligned}$$

Thus

$$ \frac{A}{ \Vert (PQ)^{\frac{1}{2}} \Vert ^{2}} \Vert f \Vert ^{2}\leq \sum _{j\in J} \Vert \Lambda _{j}f \Vert ^{2}, \quad \forall f\in \mathcal{H}. $$

For any $f\in \mathcal{H}$, it follows that

$$\begin{aligned} \sum_{j\in J} \Vert \Lambda _{j}f \Vert ^{2}&= \langle S_{\Lambda}f, f \rangle = \bigl\langle (PQ)^{-\frac{1}{2}}(PQ)^{\frac{1}{2}}S_{ \Lambda}f, f \bigr\rangle \\ &= \bigl\langle (PQ)^{\frac{1}{2}}S_{\Lambda}f, (PQ)^{-\frac{1}{2}}f \bigr\rangle \\ &= \bigl\langle S_{\Lambda}(PQ) (PQ)^{-\frac{1}{2}}f, (PQ)^{- \frac{1}{2}}f \bigr\rangle \\ &= \bigl\langle PS_{\Lambda}Q(PQ)^{-\frac{1}{2}}f, (PQ)^{-\frac{1}{2}}f \bigr\rangle \\ &\leq B \bigl\Vert (PQ)^{-\frac{1}{2}}f \bigr\Vert ^{2} \leq B \bigl\Vert (PQ)^{- \frac{1}{2}} \bigr\Vert ^{2} \Vert f \Vert ^{2}. \end{aligned}$$

Hence $\{\Lambda _{j}\}_{j\in J}$ is a g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$ with bounds $\frac{A}{ \Vert (PQ)^{\frac{1}{2}} \Vert ^{2}}$ and $B \Vert (PQ)^{-\frac{1}{2}} \Vert ^{2}$.

On the other hand, suppose that $\{\Lambda _{j}\}_{j\in J}$ is a g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$ with bounds $A_{1}$, $B_{1}$. Then

$$ \langle A_{1}f, f\rangle \leq \langle S_{\Lambda}f, f \rangle \leq \langle B_{1}f, f\rangle \quad \text{for any } f\in \mathcal{H}. $$

Since $P, Q\in \mathcal{GL}^{+}(\mathcal{H})$, by Lemma 3.1, there exist constants $c, c_{1}, C, C_{1}$ ($0< c, c_{1}, C, C_{1}<\infty$) such that

$$ cI_{\mathcal{H}}\leq P\leq CI_{\mathcal{H}},\qquad c_{1}I_{\mathcal{H}} \leq Q\leq C_{1}I_{\mathcal{H}}. $$

Using $\langle PS_{\Lambda}f, f \rangle =\langle f, S_{\Lambda}Pf \rangle = \langle f, PS_{\Lambda}f \rangle$, we get

$$ cA\leq S_{\Lambda}P=PS_{\Lambda}\leq CB. $$

Similarly, we have

$$ cc_{1}A\leq QS_{\Lambda}P\leq CC_{1}B. $$

It follows that

$$ cc_{1}A \Vert f \Vert ^{2}\leq \sum _{j\in J}\langle \Lambda _{j}Pf, \Lambda _{j}Qf\rangle \leq CC_{1}B \Vert f \Vert ^{2},\quad \forall f\in \mathcal{H}. $$

Therefore, $\{\Lambda _{j}\}_{j\in J}$ is a $(P, Q)$-controlled g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$. The proof is completed. □

Lemma 3.3

Let $P, Q\in \mathcal{GL}^{+}(\mathcal{H})$. Then $\{\Lambda _{j}\}_{j\in J}$ is a $(P, Q)$-controlled g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$ if and only if $\{\Lambda _{j}\}_{j\in J}$ is a $((QP)^{\frac{1}{2}}, (QP)^{\frac{1}{2}})$-controlled g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$.

Proof

For any $f\in \mathcal{H}$, we have

$$\begin{aligned} \sum_{j\in J}\langle \Lambda _{j}Pf, \Lambda _{j}Qf\rangle &= \biggl\langle \sum_{j\in J}Q \Lambda _{j}^{\ast}\Lambda _{j}Pf, f \biggr\rangle = \langle QS_{\Lambda}Pf, f\rangle \\ &= \langle QPS_{\Lambda}f, f \rangle = \bigl\langle (QP)^{ \frac{1}{2}}S_{\Lambda}(QP)^{\frac{1}{2}}f, f \bigr\rangle \\ &= \biggl\langle \sum_{j\in J}(QP)^{\frac{1}{2}}\Lambda _{j}^{ \ast}\Lambda _{j}(QP)^{\frac{1}{2}}f, f \biggr\rangle \\ &=\sum_{j\in J} \bigl\langle \Lambda _{j}(QP)^{\frac{1}{2}}f, \Lambda _{j} (QP)^{\frac{1}{2}}f \bigr\rangle . \end{aligned}$$

Hence, $\{\Lambda _{j}\}_{j\in J}$ is a $(P, Q)$-controlled g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$ is equivalent to

$$ A \Vert f \Vert ^{2}\leq \sum_{j\in J} \bigl\langle \Lambda _{j}(QP)^{ \frac{1}{2}}f, \Lambda _{j} (QP)^{\frac{1}{2}}f \bigr\rangle \leq B \Vert f \Vert ^{2},\quad \forall f\in \mathcal{H}, $$

where A and B are frame bounds of $\{\Lambda _{j}\}_{j\in J}$. Thus $\{\Lambda _{j}\}_{j\in J}$ is a $((QP)^{\frac{1}{2}}, (QP)^{\frac{1}{2}})$-controlled g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$. The proof is completed. □

Lemma 3.4

Let $P, Q\in \mathcal{GL}^{+}(\mathcal{H})$. Then $\{\Lambda _{j}\}_{j\in J}$ is a $(P, Q)$-controlled g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$ if and only if $\{\Lambda _{j}\}_{j\in J}$ is a QP-controlled g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$.

Proof

The proof is similar to that of Lemma 3.3. □

Lemma 3.5

Let $P, Q\in \mathcal{GL}^{+}(\mathcal{H})$. Then $\{\Lambda _{j}\}_{j\in J}$ is a $(P, Q)$-controlled g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$ if and only if $\{u_{j,k}\}_{j\in J, k\in K_{j}}$ is a $(P, Q)$-controlled g-frame for $\mathcal{H}$, where $\{u_{j,k}\}_{j\in J, k\in K_{j}}$ is the sequence induced by $\{\Lambda _{j}\}_{j\in J}$ w. r. t. $\{e_{j,k}\}_{j\in J, k\in K_{j}}$ (i.e., $u_{j,k}=\Lambda _{j}^{\ast}e_{j,k}$).

Proof

Noting that $\{e_{j,k}\}_{k\in K_{j}}$ is an o.n.b. for $\mathcal{V}_{j}$ for each $j\in J$, for any $f\in \mathcal{H}$, we have $\Lambda _{j}f\in \mathcal{V}_{j}$. It follows that

$$ \Lambda _{j}Pf=\sum_{k\in K_{j}}\langle \Lambda _{j}Pf, e_{j,k} \rangle e_{j,k}=\sum _{k\in K_{j}}\bigl\langle f, P\Lambda _{j}^{ \ast}e_{j,k} \bigr\rangle e_{j,k} $$

and

$$ \Lambda _{j}Qf=\sum_{k\in K_{j}}\langle \Lambda _{j}Qf, e_{j,k} \rangle e_{j,k}=\sum _{k\in K_{j}}\bigl\langle f, Q\Lambda _{j}^{ \ast}e_{j,k} \bigr\rangle e_{j,k}. $$

It is easy to check that

$$ \langle \Lambda _{j}Pf, \Lambda _{j}Qf\rangle =\sum _{k\in K_{j}} \bigl\langle f, P\Lambda _{j}^{\ast}e_{j,k} \bigr\rangle \bigl\langle Q\Lambda _{j}^{ \ast}e_{j,k}, f\bigr\rangle =\sum_{k\in K_{j}}\langle f, Pu_{j,k} \rangle \langle Qu_{j,k}, f\rangle . $$

Hence

$$ \sum_{j\in J}\langle \Lambda _{j}Pf, \Lambda _{j}Qf\rangle = \sum_{j\in J}\sum _{k\in K_{j}}\langle f, Pu_{j,k} \rangle \langle Qu_{j,k}, f\rangle . $$

Thus

$$ A \Vert f \Vert ^{2}\leq \sum_{j\in J} \langle \Lambda _{j}Pf, \Lambda _{j}Qf \rangle \leq B \Vert f \Vert ^{2}\quad \text{for any }f\in \mathcal{H} $$

is equivalent to

$$ A \Vert f \Vert ^{2}\leq \sum_{j\in J} \sum_{k\in K_{j}} \langle f, Pu_{j,k}\rangle \langle Qu_{j,k}, f\rangle \leq B \Vert f \Vert ^{2} \quad \text{for any }f\in \mathcal{H}. $$

The proof is completed. □

Lemma 3.6

Let $P, Q\in \mathcal{GL}^{+}(\mathcal{H})$. Then $\{\Lambda _{j}\}_{j\in J}$ is a $(P, Q)$-controlled g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$ if and only if $\{Pu_{j,k}\}_{j\in J, k\in K_{j}}$ is a $QP^{-1}$-controlled frame for $\mathcal{H}$, where $\{u_{j,k}\}_{j\in J, k\in K_{j}}$ is the sequence induced by $\{\Lambda _{j}\}_{j\in J}$ w. r. t. $\{e_{j,k}\}_{j\in J, k\in K_{j}}$ (i.e., $u_{j,k}=\Lambda _{j}^{\ast}e_{j,k}$).

Proof

From the proof of Theorem 3.5, we have

$$ \sum_{j\in J}\langle \Lambda _{j}Pf, \Lambda _{j}Qf\rangle = \sum_{j\in J}\sum _{k\in K_{j}}\bigl\langle f, P\Lambda _{j}^{ \ast}e_{j,k} \bigr\rangle \bigl\langle Q\Lambda _{j}^{\ast}e_{j,k}, f\bigr\rangle . $$

If we take $u_{j,k}=\Lambda _{j}^{\ast}e_{j,k}$, $f_{j,k}=Pu_{j,k}$, then

$$ A \Vert f \Vert ^{2}\leq \sum_{j\in J} \langle \Lambda _{j}Pf, \Lambda _{j}Qf \rangle \leq B \Vert f \Vert ^{2}\quad \text{for any }f\in \mathcal{H} $$

is equivalent to

$$ A \Vert f \Vert ^{2}\leq \sum_{j\in J} \sum_{k\in K_{j}} \langle f, Pu_{j,k}\rangle \bigl\langle QP^{-1}Pu_{j,k}, f\bigr\rangle \leq B \Vert f \Vert ^{2}\quad \text{for any }f\in \mathcal{H}. $$

The proof is completed. □

Combining Lemmas 3.2–3.6, we get Theorem 3.1.

Theorem 3.1

Let $P, Q\in \mathcal{GL}^{+}(\mathcal{H})$. Then the following are equivalent:

• (i) $\{\Lambda _{j}\}_{j\in J}$ is a $(P, Q)$-controlled g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$.

• (ii) $\{\Lambda _{j}\}_{j\in J}$ is a g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$.

• (iii) $\{\Lambda _{j}\}_{j\in J}$ is a $((QP)^{\frac{1}{2}}, (QP)^{\frac{1}{2}})$-controlled g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$.

• (iv) $\{\Lambda _{j}\}_{j\in J}$ is a QP-controlled g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$.

• (v) $\{u_{j,k}\}_{j\in J, k\in K_{j}}$ is a $(P, Q)$-controlled frame for $\mathcal{H}$, where $\{u_{j,k}\}_{j\in J, k\in K_{j}}$ is the sequence induced by $\{\Lambda _{j}\}_{j\in J}$ w. r. t. $\{e_{j,k}\}_{j\in J, k\in K_{j}}$.

• (vi) $\{Pu_{j,k}\}_{j\in J, k\in K_{j}}$ is a $QP^{-1}$-controlled frame for $\mathcal{H}$, where $\{u_{j,k}\}_{j\in J, k\in K_{j}}$ is the sequence induced by $\{\Lambda _{j}\}_{j\in J}$ w. r. t. $\{e_{j,k}\}_{j\in J, k\in K_{j}}$.

4 Controlled dual g-frames in Hilbert spaces

In this section, we introduce the notion of controlled dual frames and obtain some characterizations of the controlled dual g-frames for a given controlled g-frame by the method of operator theory.

Definition 4.1

Let $P, Q\in \mathcal{GL}^{+}(\mathcal{H})$, $\{\Lambda _{j}\}_{j\in J}$ and $\{\Gamma _{j}\}_{j\in J}$ be $(P, P)$-controlled and $(Q, Q)$-controlled g-Bessel sequences for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$, respectively. If for any $f\in \mathcal{H}$

$$ f=\sum_{j\in J}P\Lambda _{j}^{\ast} \Gamma _{j}Qf, $$

then $\{\Gamma _{j}\}_{j\in J}$ is called a $(P, Q)$-controlled dual g-frame of $\{\Lambda _{j}\}_{j\in J}$. In particular, if $Q=I_{\mathcal{H}}$, then $\{\Gamma _{j}\}_{j\in J}$ is called a P-controlled dual g-frame of $\{\Lambda _{j}\}_{j\in J}$.

Definition 4.2

Let $P, Q\in \mathcal{GL}^{+}(\mathcal{H})$, $\{\Lambda _{j}\}_{j\in J}$ and $\{\Gamma _{j}\}_{j\in J}$ be $(P, P)$-controlled and $(Q, Q)$-controlled g-Bessel sequence for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$, respectively. We define a $(P, Q)$-controlled dual g-frame operator for this pair of controlled g-Bessel sequence as follows:

$$ S_{P\Lambda \Gamma Q}f=\sum_{j\in J}P\Lambda _{j}^{\ast} \Gamma _{j}Qf,\quad \forall f\in \mathcal{H}. $$

As mentioned before, $\{\Lambda _{j}\}_{j\in J}$ and $\{\Gamma _{j}\}_{j\in J}$ are also two g-Bessel sequences. It is easy to check that $S_{P\Lambda \Gamma Q}$ is a well-defined and bounded operator, and

$$ S_{P\Lambda \Gamma Q}=T_{P\Lambda P}T_{Q\Gamma Q}^{\ast}=PT_{\Lambda}T_{ \Gamma}^{\ast}Q=PS_{\Lambda \Gamma}Q, $$

where $S_{\Lambda \Gamma}=\sum_{j\in J}\Lambda _{j}^{\ast}\Gamma _{j}$. From Definition 4.1, $\{\Gamma _{j}\}_{j\in J}$ is a $(P, Q)$-controlled dual g-frame of $\{\Lambda _{j}\}_{j\in J}$ if and only if $S_{P\Lambda \Gamma Q}=I_{\mathcal{H}}$.

Proposition 4.1

Let $P, Q\in \mathcal{GL}^{+}(\mathcal{H})$, $\{\Lambda _{j}\}_{j\in J}$ and $\{\Gamma _{j}\}_{j\in J}$ be $(P, P)$-controlled and $(Q, Q)$-controlled g-Bessel sequences with bounds $B_{\Lambda}$ and $B_{\Gamma}$, respectively. If $S_{P\Lambda \Gamma Q}$ is bounded below, then $\{\Lambda _{j}\}_{j\in J}$ and $\{\Gamma _{j}\}_{j\in J}$ are $(P, P)$-controlled and $(Q, Q)$-controlled g-frames, respectively.

Proof

Suppose that there exists a constant $\lambda >0$ such that

$$ \Vert S_{P\Lambda \Gamma Q}f \Vert \geq \lambda \Vert f \Vert \quad \text{for all }f\in \mathcal{H}. $$

By the Cauchy–Schwarz inequality, we have

$$\begin{aligned} \lambda \Vert f \Vert &\leq \Vert S_{P\Lambda \Gamma Q}f \Vert =\sup _{ \Vert g \Vert =1} \biggl\vert \biggl\langle \sum _{j\in J}P\Lambda _{j}^{\ast} \Gamma _{j}Qf, g \biggr\rangle \biggr\vert \\ &=\sup_{ \Vert g \Vert =1} \biggl\vert \sum_{j\in J} \langle \Gamma _{j}Qf, \Lambda _{j}Pg \rangle \biggr\vert \\ &\leq \sup_{ \Vert g \Vert =1} \biggl(\sum_{j\in J} \Vert \Gamma _{j}Qf \Vert ^{2} \biggr)^{\frac{1}{2}} \biggl(\sum_{j\in J} \Vert \Lambda _{j}Pg \Vert ^{2} \biggr)^{\frac{1}{2}} \\ &\leq \sqrt{B_{\Lambda }} \biggl(\sum_{j\in J} \Vert \Gamma _{j}Qf \Vert ^{2} \biggr)^{\frac{1}{2}}. \end{aligned}$$

Thus

$$ \frac{\lambda ^{2}}{B_{\Lambda}} \Vert f \Vert ^{2}\leq \sum _{j\in J} \Vert \Gamma _{j}Qf \Vert ^{2} \quad \text{for }f\in \mathcal{H}. $$

On the other hand, since

$$ S_{P\Lambda \Gamma Q}^{\ast}=(PS_{\Lambda \Gamma}Q)^{\ast} =QS_{ \Lambda \Gamma}^{\ast}P=QS_{\Gamma \Lambda}P=S_{Q\Gamma \Lambda P}, $$

then $S_{Q\Gamma \Lambda P}$ is also bounded below. Similarly, we can prove that $\{\Lambda _{j}\}_{j\in J}$ is a $(P, P)$-controlled g-frame. The proof is completed. □

Theorem 4.1

Let $P, Q\in \mathcal{GL}^{+}(\mathcal{H})$, $\{\Lambda _{j}\}_{j\in J}$ and $\{\Gamma _{j}\}_{j\in J}$ be $(P, P)$-controlled and $(Q, Q)$-controlled g-Bessel sequences for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$, respectively. Then the following conditions are equivalent:

• (i) $f=\sum_{j\in J}P\Lambda _{j}^{\ast} \Gamma _{j}Qf$, $\forall f\in \mathcal{H}$;

• (ii) $f=\sum_{j\in J}Q\Gamma _{j}^{\ast} \Lambda _{j}Pf$, $\forall f\in \mathcal{H}$;

• (iii) $\langle f, g\rangle =\sum_{j\in J} \langle \Lambda _{j}Pf, \Gamma _{j}Qg \rangle =\sum_{j\in J} \langle \Gamma _{j}Qf, \Lambda _{j}Pg \rangle$, $\forall f,g\in \mathcal{H}$;

• (iv) $\|f\|^{2}=\sum_{j\in J}\langle \Lambda _{j}Pf, \Gamma _{j}Qf \rangle =\sum_{j\in J}\langle \Gamma _{j}Qf, \Lambda _{j}Pf \rangle$, $\forall f\in \mathcal{H}$.

In case the equivalent conditions are satisfied, $\{\Lambda _{j}\}_{j\in J}$ and $\{\Gamma _{j}\}_{j\in J}$ are $(P, P)$-controlled and $(Q, Q)$-controlled g-frames, respectively.

Proof

(i)⇔(ii). Let $T_{P\Lambda P}$ be the synthesis operator of the $(P, P)$-controlled g-Bessel sequence $\{\Lambda _{j}\}_{j\in J}$ and $T_{Q\Gamma Q}$ be the synthesis operator of the $(Q, Q)$-controlled g-Bessel sequence $\{\Gamma _{j}\}_{j\in J}$. In these conditions (i) means that $T_{P\Lambda P}T_{Q\Gamma Q}^{\ast}=I_{\mathcal{H}}$, this is equivalent to $T_{Q\Gamma Q}T_{P\Lambda P}^{\ast}=I_{\mathcal{H}}$, which is identical to statement (ii). Conversely, (ii) implies (i) similarly.

(ii)⇔(iii). It is clear that (ii)⇒(iii). Next we prove (iii) implies (ii) for any $f, g\in \mathcal{H}$, $\langle f, g\rangle =\sum_{j\in J}\langle \Lambda _{j}Pf, \Gamma _{j}Qg \rangle$ shows that

$$ \biggl\langle f-\sum_{j\in J} Q\Gamma _{j}^{\ast}\Lambda _{j}Pf, g \biggr\rangle =0,\quad \forall g\in \mathcal{H}. $$

Hence (ii) is followed.

(iii)⇔(iv). (iii)⇒(iv) is obvious. To prove that (iv)⇒(iii), applying condition (iv), we have

$$\begin{aligned} \Vert f+g \Vert ^{2}&=\sum_{j\in J}\bigl\langle \Lambda _{j}P(f+g), \Gamma _{j}Q(f+g)\bigr\rangle \\ &= \sum_{j\in J}\langle \Lambda _{j}Pf+ \Lambda _{j}Pg, \Gamma _{j}Qf +\Gamma _{j}Qg \rangle \\ &=\sum_{j\in J}\langle \Lambda _{j}Pf, \Gamma _{j}Qf \rangle +\sum_{j\in J}\langle \Lambda _{j}Pf, \Gamma _{j}Qg \rangle \\ &\quad {}+\sum_{j\in J}\langle \Lambda _{j}Pg, \Gamma _{j}Qf \rangle +\sum _{j\in J}\langle \Lambda _{j}Pg, \Gamma _{j}Qg \rangle . \end{aligned}$$

Similarly,

$$\begin{aligned}& \begin{aligned} \Vert f-g \Vert ^{2}&=\sum _{j\in J}\langle \Lambda _{j}Pf, \Gamma _{j}Qf \rangle -\sum_{j\in J}\langle \Lambda _{j}Pf, \Gamma _{j}Qg \rangle \\ &\quad {}-\sum_{j\in J}\langle \Lambda _{j}Pg, \Gamma _{j}Qf \rangle +\sum _{j\in J}\langle \Lambda _{j}Pg, \Gamma _{j}Qg \rangle , \end{aligned} \\& \begin{aligned} \Vert f+\mathbf{i}g \Vert ^{2}&=\sum _{j\in J}\langle \Lambda _{j}Pf, \Gamma _{j}Qf \rangle -\mathbf{i}\sum_{j\in J} \langle \Lambda _{j}Pf, \Gamma _{j}Qg \rangle \\ &\quad {}+\mathbf{i}\sum_{j\in J}\langle \Lambda _{j}Pg, \Gamma _{j}Qf \rangle +\sum _{j\in J}\langle \Lambda _{j}Pg, \Gamma _{j}Qg \rangle , \end{aligned} \\& \begin{aligned} \Vert f-\mathbf{i}g \Vert ^{2}&=\sum _{j\in J}\langle \Lambda _{j}Pf, \Gamma _{j}Qf \rangle +\mathbf{i}\sum_{j\in J} \langle \Lambda _{j}Pf, \Gamma _{j}Qg \rangle \\ &\quad {}-\mathbf{i}\sum_{j\in J}\langle \Lambda _{j}Pg, \Gamma _{j}Qf \rangle +\sum _{j\in J}\langle \Lambda _{j}Pg, \Gamma _{j}Qg \rangle . \end{aligned} \end{aligned}$$

By polarization identity,

$$\begin{aligned} \langle f, g\rangle &=\frac{1}{4} \bigl( \Vert f+g \Vert ^{2} - \Vert f-g \Vert ^{2}+ \mathbf{i} \Vert f+ \mathbf{i}g \Vert ^{2}-\mathbf{i} \Vert f-\mathbf{i}g \Vert ^{2} \bigr) \\ &=\sum_{j\in J}\langle \Lambda _{j}Pf, \Gamma _{j}Qg \rangle . \end{aligned}$$

In case the equivalent conditions are satisfied, $S_{Q\Gamma \Lambda P}=I_{\mathcal{H}}$ implies $\|S_{Q\Gamma \Lambda P}\|=1$, by Proposition 4.1, $\{\Lambda _{j}\}_{j\in J}$ and $\{\Gamma _{j}\}_{j\in J}$ are $(P, P)$-controlled and $(Q, Q)$-controlled g-frames, respectively. The proof is completed. □

Lemma 4.1

Let $P, Q\in \mathcal{GL}^{+}(\mathcal{H})$. A sequence $\{\Lambda _{j}\}_{j\in J}$ is a $(P, Q)$-controlled g-Bessel sequence for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$ with bound B if and only if the operator

$$ T_{P\Lambda Q}: \bigoplus_{j\in J} \mathcal{V}_{j} \rightarrow \mathcal{H}, \qquad T_{P\Lambda Q}\bigl( \{f_{j}\}_{j\in J}\bigr)=\sum_{j\in J} (PQ)^{\frac{1}{2}}\Lambda _{j}^{\ast}f_{j} $$

is well defined and bounded with $\|T_{P\Lambda Q}\|\leq \sqrt{B}$.

Proof

The necessary condition follows from the definition of $(P, Q)$-controlled g-Bessel sequence. We only need to prove that the sufficient condition holds. Suppose that $T_{P\Lambda Q}$ is well defined and bounded operator with $\|T_{P\Lambda Q}\|\leq \sqrt{B}$. For any $f\in \mathcal{H}$, we have

$$\begin{aligned} \sum_{j\in J}\langle \Lambda _{j}Pf, \Lambda _{j}Qf\rangle &= \sum_{j\in J}\bigl\langle Q \Lambda _{j}^{\ast}\Lambda _{j}Pf, f \bigr\rangle = \langle QS_{\Lambda}Pf, f\rangle \\ &= \bigl\langle (QP)^{\frac{1}{2}}S_{\Lambda}(QP)^{\frac{1}{2}}f, f \bigr\rangle \\ &= \biggl\langle \sum_{j\in J} (QP)^{\frac{1}{2}} \Lambda _{j}^{ \ast}\Lambda _{j}(QP)^{\frac{1}{2}}f, f \biggr\rangle \\ &\leq \Vert T_{P\Lambda Q} \Vert \biggl(\sum_{j\in J} \bigl\Vert \Lambda _{i}(QP)^{ \frac{1}{2}}f \bigr\Vert ^{2} \biggr)^{\frac{1}{2}} \Vert f \Vert \\ &= \Vert T_{P\Lambda Q} \Vert \biggl(\sum_{j\in J} \langle \Lambda _{j}Pf, \Lambda _{j}Qf\rangle \biggr)^{\frac{1}{2}} \Vert f \Vert . \end{aligned}$$

Hence we get

$$ \sum_{j\in J}\langle \Lambda _{j}Pf, \Lambda _{j}Qf\rangle \leq \Vert T_{P\Lambda Q} \Vert ^{2} \Vert f \Vert ^{2}\leq B \Vert f \Vert ^{2}. $$

This shows that $\{\Lambda _{j}\}_{j\in J}$ is a $(P, Q)$-controlled g-Bessel sequence for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$ with bound B. The proof is completed. □

Theorem 4.2

Let $P, Q\in \mathcal{GL}^{+}(\mathcal{H})$, $\{\Lambda _{j}\}_{j\in J}$ be a $(P, P)$-controlled g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$ with the synthesis operator $T_{P\Lambda P}$. Then a $(Q, Q)$-controlled g-frame $\{\Gamma _{j}\}_{j\in J}$ is a $(P, Q)$-controlled dual g-frame of $\{\Lambda _{j}\}_{j\in J}$ if and only if

$$ Q\Gamma _{j}^{\ast}e_{j,k}=U(e_{j,k}\delta _{j}),\quad j\in J, k\in K_{j}, $$

where $U: \bigoplus_{j\in J}\mathcal{V}_{j}\rightarrow \mathcal{H}$ is a bounded left-inverse of $T_{P\Lambda P}^{\ast}$.

Proof

If $\{g_{j}\}_{j\in J}\in \bigoplus_{j\in J}\mathcal{V}_{j}$, then

$$ \{g_{j}\}_{j\in J}=\sum_{j\in J}g_{j} \delta _{j}=\sum_{j\in J}\sum _{k\in K_{j}}\langle g_{j}, e_{j,k} \rangle e_{j,k}\delta _{j}. $$

Roughly speaking, $\{e_{j,k}\delta _{j}\}_{j\in J,k\in K_{j}}$ is an o. n. b. of $\bigoplus_{j\in J}\mathcal{V}_{j}$. If there exist $U: \bigoplus_{j\in J}\mathcal{V}_{j}\rightarrow \mathcal{H}$ is a bounded left-inverse of $T_{P\Lambda P}^{\ast}$ such that

$$ Q\Gamma _{j}^{\ast}e_{j,k}=U(e_{j,k}\delta _{j}),\quad j\in J, k\in K_{j}. $$

By Lemma 4.1, $\{\Gamma _{j}\}_{j\in J}$ is a $(Q, Q)$-controlled g-Bessel sequence for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$. For any $f\in \mathcal{H}$, we have

$$\begin{aligned} f&=UT_{P\Lambda P}^{\ast}f=U \biggl(\sum_{j\in J} \sum_{k\in K_{j}}\langle \Lambda _{j}Pf, e_{j,k}\rangle e_{j,k} \delta _{j} \biggr) \\ &= \sum_{j\in J}\sum_{k\in K_{j}} \bigl\langle f, P \Lambda _{j}^{\ast}e_{j,k}\bigr\rangle U(e_{j,k}\delta _{j}) \\ &= \sum_{j\in J}\sum_{k\in K_{j}} \langle f, Pu_{j,k} \rangle Q\Gamma _{j}^{\ast}e_{j,k} \\ &= \sum_{j\in J}Q\Gamma _{j}^{\ast} \biggl(\sum_{k \in K_{j}} \langle Pf, u_{j,k}\rangle e_{j,k} \biggr)=\sum_{j \in J}Q\Gamma _{j}^{\ast}\Lambda _{j}Pf, \end{aligned}$$

where $u_{j,k}=\Lambda _{j}^{\ast}e_{j,k}$. By the definition of controlled dual g-frame, $\{\Gamma _{j}\}_{j\in J}$ is a $(P, Q)$-controlled dual g-frame of $\{\Lambda _{j}\}_{j\in J}$.

On the other hand, suppose that a $(Q, Q)$-controlled g-frame $\{\Gamma _{j}\}_{j\in J}$ is a $(P, Q)$-controlled dual g-frame of $\{\Lambda _{j}\}_{j\in J}$. For any $f\in \mathcal{H}$, we have

$$ f=\sum_{j\in J}P\Lambda _{j}^{\ast} \Gamma _{j}Qf=\sum_{j\in J}Q\Gamma _{j}^{\ast}\Lambda _{j}Pf, $$

that is, $T_{Q\Gamma Q}T_{P\Lambda P}^{\ast}=I_{\mathcal{H}}$. Let $U=T_{Q\Gamma Q}$, then $U: \bigoplus_{j\in J}\mathcal{V}_{j}\rightarrow \mathcal{H}$ is a bounded left-inverse of $T_{P\Lambda P}^{\ast}$. A calculation as above shows that

$$ \sum_{j\in J}\sum_{k\in K_{j}} \langle f, Pu_{j,k} \rangle Q\Gamma _{j}^{\ast}e_{j,k}=f= \sum_{j\in J}\sum_{k\in K_{j}} \langle f, Pu_{j,k}\rangle U(e_{j,k}\delta _{j}), \quad \forall f\in \mathcal{H}. $$

Combining this with the fact $\{e_{j,k}\}_{k\in K_{j}}$ is an o. n. b. of $\mathcal{V}_{j}$, we have

$$ Q\Gamma _{j}^{\ast}e_{j,k}=U(e_{j,k}\delta _{j}),\quad j\in J, k\in K_{j}. $$

The proof is completed. □

Theorem 4.3

Let $P\in \mathcal{GL}^{+}(\mathcal{H})$, $\{\Lambda _{j}\}_{j\in J}$ be a $(P, P)$-controlled g-frame for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$ with the synthesis operator and frame operator $T_{P\Lambda P}$ and $S_{P\Lambda P}$, respectively. Then $\{\Gamma _{j}\in L(\mathcal{H}, \mathcal{V}_{j})\}_{j\in J}$ is a P-controlled dual g-frame of $\{\Lambda _{j}\}_{j\in J}$ if and only if

$$ \Gamma _{j}f=(Tf)_{j}+\Lambda _{j}S_{P\Lambda P}^{-1}Pf, \quad j\in J, f \in \mathcal{H}, $$

where $T: \mathcal{H}\rightarrow \bigoplus_{j\in J}\mathcal{V}_{j}$ is a bounded linear operator satisfying $T_{P\Lambda P}T=0$.

Proof

If $T: \mathcal{H}\rightarrow \bigoplus_{j\in J}\mathcal{V}_{j}$ is a bounded linear operator satisfying $T_{P\Lambda P}T=0$, then $\{\Gamma _{j}\in L(\mathcal{H}, \mathcal{V}_{j})\}_{j\in J}$ is a g-Bessel sequence for $\mathcal{H}$ w. r. t. $\{\mathcal{V}_{j}\}_{j\in J}$. In fact, for any $f\in \mathcal{H}$, we have

$$\begin{aligned} \sum_{j\in J} \Vert \Gamma _{j}f \Vert ^{2}&=\sum_{j\in J} \bigl\Vert (Tf)_{j}+ \Lambda _{j}S_{P\Lambda P}^{-1}Pf \bigr\Vert ^{2} \\ &\leq 2 \biggl(\sum_{j\in J} \bigl\Vert \Lambda _{j}S_{P\Lambda P}^{-1}Pf \bigr\Vert ^{2}+ \Vert Tf \Vert ^{2} \biggr) \\ &\leq 2 \bigl(B \bigl\Vert S_{P\Lambda P}^{-1}P \bigr\Vert ^{2}+ \Vert T \Vert ^{2} \bigr) \Vert f \Vert ^{2}, \end{aligned}$$

where B is the upper bound of $\{\Lambda _{j}\}_{j\in J}$. Furthermore,

$$\begin{aligned} \sum_{j\in J}P\Lambda _{j}^{\ast} \Gamma _{j}f&=\sum_{j \in J}P\Lambda _{j}^{\ast}\bigl((Tf)_{j}+\Lambda _{j}S_{P\Lambda P}^{-1}Pf\bigr) \\ &=T_{P\Lambda T}Tf+\sum_{j\in J}P\Lambda _{j}^{\ast}\Lambda _{j}S_{P \Lambda P}^{-1}Pf=f. \end{aligned}$$

Thus $\{\Gamma _{j}\in L(\mathcal{H}, \mathcal{V}_{j})\}_{j\in J}$ is a P-controlled dual g-frame of $\{\Lambda _{j}\}_{j\in J}$.

Now we prove the converse. Assume that $\{\Gamma _{j}\in L(\mathcal{H}, \mathcal{V}_{j})\}_{j\in J}$ is a P-controlled dual g-frame of $\{\Lambda _{j}\}_{j\in J}$. Define the operator T as follows:

$$ T: \mathcal{H}\rightarrow \bigoplus_{j\in J} \mathcal{V}_{j},\qquad f \mapsto Sf\quad (\forall f\in \mathcal{H}) $$

satisfying

$$ \Gamma _{j}f=(Tf)_{j}+\Lambda _{j}S_{P\Lambda P}^{-1}Pf, \quad j\in J. $$

For any $f\in \mathcal{H}$, we have

$$\begin{aligned} \Vert Tf \Vert ^{2}&=\sum_{j\in J} \bigl\Vert \Gamma _{j}f-\Lambda _{j}S_{P \Lambda P}^{-1}Pf \bigr\Vert ^{2} \\ &\leq \sum_{j\in J} \Vert \Gamma _{j}f \Vert ^{2}+\sum_{j\in J} \bigl\Vert \Lambda _{j}S_{P\Lambda P}^{-1}Pf \bigr\Vert ^{2}+2 \biggl(\sum_{j \in J} \Vert \Gamma _{j}f \Vert ^{2} \biggr)^{\frac{1}{2}} \biggl(\sum _{j \in J} \bigl\Vert \Lambda _{j}S_{P\Lambda P}^{-1}Pf \bigr\Vert ^{2} \biggr)^{\frac{1}{2}} \\ &\leq \bigl(B_{1}+A^{-1}+2\sqrt{B_{1}A^{-1}} \bigr) \Vert f \Vert ^{2}, \end{aligned}$$

where $B_{1}$ is the frame upper bound of $\{\Gamma _{j}\}_{j\in J}$, A is the frame lower bound of $\{\Lambda _{j}\}_{j\in J}$. Thus T is a linear bounded operator. Moreover, for any $f, g\in \mathcal{H}$, we have

$$\begin{aligned} \langle T_{P\Lambda P}Tf, g\rangle &=\sum_{j\in J} \bigl\langle P \Lambda _{j}^{\ast}Tf, g\bigr\rangle = \sum _{j\in J}\bigl\langle P \Lambda _{j}^{\ast} \bigl(\Gamma _{j}f-\Lambda _{j}S_{P\Lambda P}^{-1}Pf \bigr), g \bigr\rangle \\ &=\sum_{j\in J}\bigl\langle P\Lambda _{j}^{\ast}\Gamma _{j}f, g \bigr\rangle -\sum _{j\in J}\bigl\langle P\Lambda _{j}^{\ast} \Lambda _{j}S_{P \Lambda P}^{-1}Pf, g\bigr\rangle \\ &=\langle f, g\rangle -\langle f, g\rangle =0. \end{aligned}$$

That is, $T_{P\Lambda P}T=0$. The proof is completed. □