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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, 1720]. 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:

  1. (i)

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

  2. (ii)

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

    $$ c \Vert f \Vert ^{2}\leq \bigl\Vert T^{\frac{1}{2}}f \bigr\Vert ^{2}\leq C \Vert f \Vert ^{2}. $$
  3. (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 3.23.6, we get Theorem 3.1.

Theorem 3.1

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

  1. (i)

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

  2. (ii)

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

  3. (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. \(\{\mathcal{V}_{j}\}_{j\in J}\).

  4. (iv)

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

  5. (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. \(\{e_{j,k}\}_{j\in J, k\in K_{j}}\).

  6. (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:

  1. (i)

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

  2. (ii)

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

  3. (iii)

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

  4. (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. □

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Acknowledgements

The authors thank the referees for their comments which greatly improve the readability of this article.

Funding

This paper is supported by the Science and Technology Research Project of Henan Province (No. 222102210335) and the Educational Commission of Henan province of China (No. 20A110036).

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Conceptualization, HML, YLF,YT; Formal analysis, TY, HML; Validation, HML, YLF,YT; Writing—original draft, HML, YLF,YT. All the authors contributed equally and they read and approved the final manuscript for publication.

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Correspondence to Yu Tian.

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Liu, HM., Fu, YL. & Tian, Y. Controlled g-frames and dual g-frames in Hilbert spaces. J Inequal Appl 2023, 64 (2023). https://doi.org/10.1186/s13660-023-02972-8

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