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. □