1 Introduction

The notion of frame dates back to Gabor [1] (1946) and Duffin and Schaeffer [2] (1952). Gabor [1] proposed the idea of decomposing a general signal in terms of elementary signals, and Duffin and Schaeffer [2] abstracted “these elementary signals” as the notion of frame. However, the frame theory had not attracted much attention until the celebrated work by Daubechies, Crossman, and Meyer [3] in 1986. So far, the theory of frame has seen great achievements in pure mathematics, science, and engineering ([4–13]). In 2006, Sun [14] introduced a generalized frame (simply g-frame), which covers all other generalizations of frames, for example, fusion frames [15], bounded quasiprojectors [16], and so on. Now, the research of g-frames has obtained many results [17–19]. This paper addresses approximately dual g-frames in Hilbert spaces.

Recall that a sequence \(\{f_{i}\}_{i\in I}\) in a separable Hilbert space H is a frame if

$$A_{1} \Vert f \Vert ^{2}\leq\sum _{i\in I} \bigl\vert \langle f, f_{n}\rangle \bigr\vert ^{2}\leq B_{1} \Vert f \Vert ^{2} $$

for all \(f\in\mathcal{H}\) and some positive constants \(A_{1}\), \(B_{1}\). Given a frame \(\{f_{i}\}_{i\in I}\), another frame \(\{h_{i}\}_{i\in I}\) is said to be a dual frame of \(\{f_{i}\}_{i\in I}\) if

$$ f=\sum_{i\in I}\langle f, f_{i}\rangle h_{i}, \quad\forall f\in{H}, $$

or, equivalently,

$$ f=\sum_{i\in I}\langle f, h_{i}\rangle f_{i}, \quad\forall f\in{H}. $$

To find the dual frames for a general frame is a fundamental problem in the frame theory. Usually, it is not easy due to involving complicated computation. In 2010, Christensen [20] introduced the notion of approximately dual frames. Bessel sequences \(\{f_{i}\}_{i\in I}\) and \(\{h_{i}\}_{i\in I}\) in a separable Hilbert space \(\mathcal{H}\) are said to be approximately dual frames if

$$\biggl\Vert f-\sum_{i\in I}\langle f, h_{i}\rangle f_{i} \biggr\Vert \leq \Vert f \Vert ,\quad \forall f\in{H}, $$

or

$$\biggl\Vert f-\sum_{i\in I}\langle f, f_{i}\rangle h_{i} \biggr\Vert \leq \Vert f \Vert ,\quad \forall f\in{H}. $$

In 2014, Khosravi et al. [21] first introduced the notion of approximately dual g-frames, which generalize the usual approximately dual frames. They proved that a pair of operator sequences form approximately dual frames if and only if their induced sequences form a pair of approximately dual g-frames. They also obtained some important properties and applications of approximately dual frames. Later, many results on approximately dual g-frames were obtained (see [22, 23]).

Motivated by [21], in this paper, we focus on the characterization and stability of approximately dual g-frames and their connection with dual g-frames. Sect. 2 is an auxiliary one, where we recall some basic notions, properties, and some related results. In Sect. 3, we establish a characterization of approximately dual g-frames and discuss some properties of approximately dual (dual) g-frames. In Sect. 4, we give some stability results of approximately dual g-frames, which cover the results obtained by other authors.

2 Preliminaries

We begin with some basic notions and results of g-frames. See [14, 17, 18] for details.

Given separable Hilbert spaces H and V, let \(\{V_{j}:j\in J\}\) be a sequence of closed subspaces of V with J being a subset of integers \(\mathbb {Z}\). The identity operator on H is denoted by \(I_{H}\). The set of all bounded linear operators from H into \(V_{j}\) is denoted by \(L(H, V_{j})\). Define

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

Then \(\bigoplus_{j\in J}V_{j}\) is a Hilbert space under the inner product

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

Suppose \(\{e_{j,k}\}_{k\in K_{j}}\) is an orthonormal basis (simply o.n.b.) for \(V_{j}\), where \(K_{j}\subset\mathbb{Z}\), \(j\in J\). Define \(\tilde{e}_{j,k}=\{\delta_{j,i}e_{i,k}\}_{i\in J}\), where δ is the Kronecker symbol. Then \(\{\tilde{e}_{j,k}\}_{j\in J,k\in K_{j}}\) is an o.n.b. for \(\bigoplus_{j\in J}V_{j}\) (see [17]).

Definition 2.1

([14])

A sequence \(\{\Lambda_{j}\in L(H, V_{j})\}_{j\in J}\) is called a g-frame for H with respect to (w.r.t.) \(\{V_{j}\}_{j\in J}\) 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 \(f\in H\) 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, then \(\{\Lambda_{j}\}_{j\in J}\) is called a g-Bessel sequence for H w.r.t. \(\{V_{j}\}_{j\in J}\) with bound B. If \(A=B=\lambda\), then \(\{\Lambda_{j}\}_{j\in J}\) is called a λ-tight g-frame. In addition, if \(\lambda=1\), then \(\{ \Lambda_{j}\}_{j\in J}\) is called a Parsevel g-frame.

For a g-Bessel sequence \(\{\Lambda_{j}\}_{j\in J}\) with bound B, the operator

$$T_{\Lambda}: \bigoplus_{j\in J}V_{j} \rightarrow H,\qquad T_{\Lambda}F=\sum_{j\in J} \Lambda_{j}^{\ast}f_{j}, \quad\forall F=\{ f_{j}\}_{j\in J}\in\bigoplus_{j\in J}V_{j}, $$

is well-defined, and its adjoint is given by

$$T_{\Lambda}^{\ast}: H \rightarrow\bigoplus _{j\in J}V_{j},\qquad T_{\Lambda}^{\ast}f=\{ \Lambda_{j}f\}_{j\in J},\quad \forall f\in H. $$

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

$$S_{\Lambda}: H\rightarrow H,\qquad S_{\Lambda}f=\sum _{j\in J}\Lambda _{j}^{\ast} \Lambda_{j}f, \quad\forall f\in H, $$

is called a g-frame operator of \(\{\Lambda_{j}\}_{j\in J}\). It is bounded, invertible, self-adjoint, and 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 also a g-frame for H w.r.t. \(\{V_{j}\}_{j\in J}\) with the g-frame operator \(S_{\Lambda}^{-1}\) and frame bounds \(\frac{1}{B}\) and \(\frac {1}{A}\). \(\{\tilde{\Lambda}_{j}\}_{j\in J}\) is called thebcanonical dual g-frame of \(\{\Lambda_{j}\}_{j\in J}\) (see [14]).

Definition 2.2

([14])

Let \(\{\Lambda_{j}\}_{j\in J}\) be a g-frame for H w.r.t. \(\{V_{j}\}_{j\in J}\). A g-frame \(\{\Gamma_{j}\}_{j\in J}\) is called an alternate dual g-frame for \(\{\Lambda_{j}\}_{j\in J}\) if

$$f=\sum_{j\in J}\Gamma_{j}^{\ast} \Lambda_{j}f,\quad \forall f\in H. $$

Moreover, \(\{\Lambda_{j}\}_{j\in J}\) is also an alternate dual g-frame for \(\{\Gamma_{j}\}_{j\in J}\), that is,

$$f=\sum_{j\in J}\Lambda_{j}^{\ast} \Gamma_{j}f, \quad\forall f\in H. $$

Definition 2.3

([20])

Let \(\{f_{j}\}_{j\in J}\) and \(\{g_{j}\}_{j\in J}\) be two Bessel sequences for H with their respective synthesis operators \(T_{f}\) and \(T_{g}\). We say that\(\{f_{j}\}_{j\in J}\) and \(\{g_{j}\}_{j\in J}\) are approximately dual frames if \(\|I_{H}-T_{f}T^{*}_{g}\|<1\) or \(\|I_{H}-T_{g}T^{*}_{f}\|<1\).

It is clear that the operator \(T_{f}T^{*}_{g}\) is invertible.

Definition 2.4

([21])

Let \(\{\Lambda_{j}\}_{j\in J}\) and \(\{\Gamma_{j}\}_{j\in J}\) be two g-Bessel sequences for H w.r.t. \(\{V_{j}\}_{j\in J}\) with their respective synthesis operators \(T_{\Lambda}\) and \(T_{\Gamma}\). Then \(\{\Lambda_{j}\}_{j\in J}\) and \(\{\Gamma_{j}\} _{j\in J}\) are approximately dual g-frames if \(\|I_{H}-T_{\Lambda}T^{*}_{\Gamma}\| <1\) or \(\|I_{H}-T_{\Gamma}T^{*}_{\Lambda}\|<1\).

3 Dual and approximately dual g-frames

This section focuses on the connection between approximately dual g-frames and dual g-frames and on a characterization of approximately dual g-frames.

Lemma 3.1

([19])

Let \(\{\Lambda_{j}\}_{j\in J}\) and \(\{\Gamma_{j}\}_{j\in J}\) be two g-Bessel sequences for H w.r.t. \(\{V_{j}\}_{j\in J}\). Then the following are equivalent:

  1. (i)

    \(f=\sum_{j\in J}\Gamma_{j}^{\ast}\Lambda_{j}f\), \(\forall f\in H\).

  2. (ii)

    \(f=\sum_{j\in J}\Lambda_{j}^{\ast}\Gamma_{j}f\), \(\forall f\in H\).

  3. (iii)

    \(\langle f, g\rangle=\sum_{j\in J}\langle\Lambda_{j}f, \Gamma_{j}g \rangle\), \(\forall f,g\in H\).

In case the equivalent conditions are satisfied, \(\{\Lambda_{j}\} _{j\in J}\) and \(\{\Gamma_{j}\}_{j\in J}\) are dual g-frames for H w.r.t. \(\{V_{j}\}_{j\in J}\).

Lemma 3.2

([14])

Let \(\Lambda_{j}\in L(H, V_{j})\) for every \(j\in J\), and let \(\{e_{j,k}\}_{k\in K_{j}}\) be an o.n.b. for \(V_{j}\). If \(u_{j,k}\) is defined by \(u_{j,k}=\Lambda_{j}^{\ast}e_{j,k}\), then \(\{\Lambda_{j}\}_{j\in J}\) is a g-frame (g-Bessel sequence) for H if and only if \(\{u_{j,k}\}_{j\in J,k\in K_{j}}\) is a frame (Bessel sequence) for H.

The following two theorems give a method to construct new dual g-frames (approximately dual g-frames) from given dual g-fromes.

Theorem 3.1

Let \(\{\Lambda_{j}\}_{j\in J}\) and \(\{ \Gamma_{j}\}_{j\in J}\) be dual g-frames for H w.r.t. \(\{V_{j}\} _{j\in J}\), and let \(O_{1}\) and \(O_{2}\) be two bounded operators on H such that \(O_{2}O_{1}^{\ast}=I_{H}\) (\(\|I_{H}-O_{2}O_{1}^{\ast}\|<1\)). Then \(\{\Lambda_{j}O_{1}\}_{j\in J}\) and \(\{\Gamma_{j}O_{2}\}_{j\in J}\) are dual g-frames (approximately dual g-frames) for H w.r.t. \(\{ V_{j}\}_{j\in J}\).

Proof

By a standard argument, \(\{\Lambda_{j}\}_{j\in J}\) is a g-Bessel sequence with synthesis operator \(T_{\Lambda}\). Since \(O_{1}\) is a bounded operator on H, we see that \(\{\Lambda_{j}O_{1}\}_{j\in J}\) is a g-Bessel sequence with synthesis operator \(T_{O\Lambda }=O_{1}T_{\Lambda}\). Similarly, \(\{\Gamma_{j}O_{2}\}_{j\in J}\) is also a g-Bessel sequence with synthesis operator \(T_{O\Gamma}=O_{2}T_{\Gamma}\). By Lemma 3.1 we have

$$\begin{gathered} T_{O\Gamma}T_{O\Lambda}^{\ast}f=O_{2}T_{\Gamma}T_{\Lambda}^{\ast }O_{1}^{\ast}f=O_{2}O_{1}^{\ast}f=f \\ \bigl( \bigl\Vert I_{H}-T_{O\Gamma}T_{O\Lambda}^{\ast} \bigr\Vert = \bigl\Vert I_{H}-O_{2}T_{\Gamma}T_{\Lambda}^{\ast}O_{1}^{\ast} \bigr\Vert = \bigl\Vert I_{H}-O_{2}O_{1}^{\ast} \bigr\Vert < 1\bigr)\end{gathered} $$

for all \(f\in H\). □

Corollary 3.1

Let \(\{\Lambda_{j}\}_{j\in J}\) and \(\{\Gamma_{j}\} _{j\in J}\) be dual g-frames for H w.r.t. \(\{V_{j}\}_{j\in J}\), and let T be a unitary operator on H. Then \(\{\Lambda_{j}T\}_{j\in J}\) and \(\{\Gamma_{j}T\}_{j\in J}\) are dual g-frames (approximately dual g-frames) for H w.r.t. \(\{V_{j}\}_{j\in J}\).

Theorem 3.2

Assume that \(\{\Lambda_{j}\}_{j\in J}\) and \(\{\Gamma _{j}\}_{j\in J}\) are dual g-frames for H w.r.t. \(\{V_{j}\}_{j\in J}\), and let \(\{\Lambda_{j}\}_{j\in J}\) and \(\{\Delta_{j}\}_{j\in J}\) also be dual g-frames for H w.r.t. \(\{V_{j}\}_{j\in J}\). Then for any \(\alpha\in\mathbb{C}\), \(\{\Lambda_{j}\}_{j\in J}\) and \(\{\alpha \Gamma_{j}+(1-\alpha)\Delta_{j}\}_{j\in J}\) are dual g-frames for H w.r.t. \(\{V_{j}\}_{j\in J}\).

Proof

By a standard argument, \(\{\alpha\Gamma_{j}+(1-\alpha )\Delta_{j}\}_{j\in J}\) is a g-Bessel sequence for H w.r.t. \(\{ V_{j}\}_{j\in J}\). By Lemma 3.1 we have

$$\begin{aligned} \sum_{j\in J}\bigl\langle \Lambda_{j}f, \bigl(\alpha\Gamma_{j}+(1-\alpha )\Delta_{j}\bigr)g \bigr\rangle &= \sum_{j\in J}\langle\Lambda_{j}f, \alpha\Gamma_{j}g \rangle +\sum_{j\in J}\bigl\langle \Lambda_{j}f, (1-\alpha)\Delta_{j}g \bigr\rangle \\ &=\bar{\alpha}\sum_{j\in J}\langle \Lambda_{j}f, \Gamma_{j}g \rangle +(1-\bar{\alpha})\sum _{j\in J}\langle\Lambda_{j}f, \Delta_{j}g \rangle \\ &=\bar{\alpha}\langle f, g\rangle+(1-\bar{\alpha})\langle f, g\rangle\\ &= \langle f, g\rangle \end{aligned}$$

for all \(f, g\in H\). □

Obviously, if \(\{\Lambda_{j}\}_{j\in J}\) and \(\{\Gamma_{j}\}_{j\in J}\) are dual g-frames for H w.r.t. \(\{V_{j}\}_{j\in J}\), then \(\{ \Lambda_{j}\}_{j\in J}\) and \(\{\Gamma_{j}\}_{j\in J}\) are approximately dual g-frames for H w.r.t. \(\{V_{j}\}_{j\in J}\). However, the converse is not true in general. The following theorem gives a sufficient condition for approximately dual g-frames to be dual g-frames.

Theorem 3.3

Let \(\{\Lambda_{j}\}_{j\in J}\) and \(\{ \Gamma_{j}\}_{j\in J}\) be approximately dual g-frames for H w.r.t. \(\{V_{j}\}_{j\in J}\) with synthesis operators \(T_{\Lambda}\) and \(T_{\Gamma}\), respectively. Then \(T_{\Lambda}T_{\Gamma}^{\ast}\) is invertible; furthermore, the sequences \(\{\Lambda_{j}\}_{j\in J}\) and \(\{(T_{\Lambda}T_{\Gamma}^{\ast})^{-1}\Gamma_{j}\}_{j\in J}\) are dual g-frames.

Proof

Since \(\{\Lambda_{j}\}_{j\in J}\) and \(\{\Gamma_{j}\}_{j\in J}\) are approximately dual g-frames for H w.r.t. \(\{V_{j}\}_{j\in J}\), we have \(\|I_{U}-T_{\Lambda}T_{\Gamma}^{\ast}\|<1\), and thus \(T_{\Lambda}T_{\Gamma}^{\ast}\) is invertible on H. By Lemma 3.1 we have

$$\begin{aligned} \langle f, g\rangle&= \bigl\langle \bigl(T_{\Lambda}T_{\Gamma}^{\ast}\bigr) \bigl(T_{\Lambda}T_{\Gamma}^{\ast}\bigr)^{-1}f, g\bigr\rangle \\ &=\bigl\langle T_{\Gamma}^{\ast}\bigl(T_{\Lambda}T_{\Gamma}^{\ast}\bigr)^{-1}f, T_{\Lambda}^{\ast}g\bigr\rangle \\ &=\sum_{j\in J}\bigl\langle \Gamma_{j}\bigl(T_{\Lambda}T_{\Gamma}^{\ast}\bigr)^{-1}f, \Lambda_{j}g \bigr\rangle \end{aligned} $$

for all \(f, g\in H\). □

For Theorem 3.3, a natural question is whether a g-frame always corresponds to an approximately dual g-frame. The following theorem gives an affirmative answer.

Theorem 3.4

Let \(\{\Lambda_{j}\}_{j\in J}\) be a g-frame for H w.r.t. \(\{V_{j}\}_{j\in J}\) with the synthesis operator \(T_{\Lambda}\) and frame bounds A and B. Then \(\{B^{-1}\Lambda_{j}\} _{j\in J}\) is an approximately dual g-frame of \(\{\Lambda_{j}\}_{j\in J}\).

Proof

Note that \(\{\Lambda_{j}\}_{j\in J}\) is a g-frame for H w.r.t. \(\{V_{j}\}_{j\in J}\) and \(T_{\Lambda}\) is its synthesis operator. So \(\{B^{-1}\Lambda_{j}\}_{j\in J}\) is also g-frame with synthesis operator \(B^{-1}T_{\Lambda}\), and

$$\begin{aligned} \bigl\Vert I_{H}-B^{-1}T_{\Lambda}T_{\Lambda}^{\ast}\bigr\Vert &=\sup_{ \Vert f \Vert =1} \bigl\vert \bigl\langle \bigl(I_{H}-B^{-1}T_{\Lambda}T_{\Lambda}^{\ast}\bigr)f, f\bigr\rangle \bigr\vert \\&\leq \frac{B-A}{B}< 1.\end{aligned} $$

It follows that \(\{B^{-1}\Lambda_{j}\}_{j\in J}\) is an approximately dual g-frame of \(\{\Lambda_{j}\}_{j\in J}\). □

From Theorem 3.4 we know that every g-frame has at least an approximately dual g-frame. Next, we characterize all approximately dual g-frames for a given g-frame. For this purpose, we need to establish some lemmas.

Lemma 3.3

Let \(\{\Lambda_{j}\}_{j\in J}\) be a g-frame for H w.r.t. \(\{V_{j}\}_{j\in J}\), let \(T_{\Lambda}\) be its synthesis operator, and let \(\{\tilde{e}_{j,k}\}_{j\in J,k\in K_{j}}\) be an o.n.b. for \(\bigoplus_{j\in J}V_{j}\). Then \(\{\Gamma_{j}\}_{j\in J}\) and \(\{\Lambda_{j}\}_{j\in J}\) are approximately dual g-frames if and only if \(\Gamma_{j}^{\ast}e_{j,k}=T\tilde{e}_{j,k}\) (\(\forall j\in J\), \(k\in K_{j}\)), where \(T: \bigoplus_{j\in J}V_{j}\rightarrow H\) is a linear bounded operator such that \(\|I_{H}-TT_{\Lambda}^{\ast}\|<1\).

Proof

Necessity. Suppose \(\{\Gamma_{j}\}_{j\in J}\) is an approximately dual g-frame of \(\{\Lambda_{j}\}_{j\in J}\). Then \(\{ \Gamma_{j}\}_{j\in J}\) is a g-frame, and \(\|I_{H}-T_{\Gamma}T_{\Lambda}^{\ast}\|<1\), where \(T_{\Gamma}\) is the synthesis operator of \(\{\Gamma _{j}\}_{j\in J}\). Notice that

$$T_{\Gamma}\tilde{e}_{j,k}=T_{\Gamma}\bigl(\{ \delta_{j,i}e_{i,k}\}_{i\in J}\bigr)= \sum _{i\in J}\Gamma_{j}^{\ast}\delta_{j,i}e_{i,k}= \Gamma_{j}^{\ast }e_{j,k}. $$

Denote \(T=T_{\Gamma}\). Then \(T: \bigoplus_{j\in J}V_{j}\rightarrow H\) is a linear bounded operator satisfying \({\|I_{H}-TT_{\Lambda}^{\ast}\|<1}\) and \(\Gamma_{j}^{\ast}e_{j,k}=T\tilde{e}_{j,k}\) for \(j\in J\), \(k\in K_{j}\).

Next, we prove the converse. Suppose \(T: \bigoplus_{j\in J}V_{j}\rightarrow H\) is a linear bounded operator satisfying \(\| I_{H}-TT_{\Lambda}^{\ast}\|<1\) and \(\Gamma_{j}^{\ast}e_{j,k}=T\tilde {e}_{j,k}\) for \(j\in J\), \(k\in K_{j}\). Then

$$\begin{aligned} TT_{\Lambda}^{\ast}f& =T \bigl(\{\Lambda_{j}f \}_{j\in J} \bigr) \\ &=T \biggl(\sum_{j\in J}\sum _{k\in K_{j}}\langle\Lambda_{j}f, e_{j,k}\rangle \tilde{e}_{j,k} \biggr) \\ &=\sum_{j\in J}\sum_{k\in K_{j}} \langle\Lambda_{j}f, e_{j,k}\rangle T\tilde{e}_{j,k} \\ &=\sum_{j\in J}\sum_{k\in K_{j}} \langle\Lambda_{j}f, e_{j,k}\rangle \Gamma_{j}^{\ast} e_{j,k} \\ & =\sum_{j\in J}\Gamma_{j}^{\ast}\sum _{k\in K_{j}}\langle\Lambda _{j}f, e_{j,k}\rangle e_{j,k} \\ & =\sum_{j\in J}\Gamma_{j}^{\ast} \Lambda_{j}f\end{aligned} $$

for \(f\in H\). Since \(\{\tilde{e}_{j,k}\}_{j\in J,k\in K_{j}}\) is an o.n.b. for \(\bigoplus_{j\in J}V_{j}\), we have that \(\{T\tilde{e}_{j,k}\}_{j\in J,k\in K_{j}}\) is a Bessel sequence for H. Let \(u_{j,k}=T\tilde {e}_{j,k}\). Then \(u_{j,k}=\Gamma_{j}^{\ast}e_{j,k}\). By Lemma 3.2 \(\{\Gamma_{j}\}_{j\in J}\) is a g-Bessel sequence for H w.r.t. \(\{V_{j}\}_{j\in J}\). Let \(T_{\Gamma}\) be the synthesis operator of \(\{ \Gamma_{j}\}_{j\in J}\). Then \(T=T_{\Gamma}\) and \(\|I_{H}-T_{\Gamma}T_{\Lambda}^{\ast}\|<1\), and hence \(\{\Gamma_{j}\}_{j\in J}\) and \(\{ \Lambda_{j}\}_{j\in J}\) are approximately dual g-frames. □

From Lemma 3.3 we know that T is very important. The following lemma gives an explicit expression of T in Lemma 3.3.

Lemma 3.4

Let \(\{\Lambda_{j}\}_{j\in J}\) be a g-frame for H w.r.t. \(\{V_{j}\}_{j\in J}\) with the synthesis operator \(T_{\Lambda}\) and the frame operator \(S_{\Lambda}\). Then \(\| I_{H}-TT_{\Lambda}^{\ast}\|<1\) (\(T: \bigoplus_{j\in J}V_{j}\rightarrow H\)) if and only if \(T=S_{\Lambda}^{-1}T_{\Lambda}+W(I-T_{\Lambda}^{\ast}QS_{\Lambda}^{-1}T_{\Lambda})\), where I is the identity operator on \(\bigoplus_{j\in J}V_{j}\), and \(W: \bigoplus_{j\in J}V_{j}\rightarrow H\) and \(Q: H\rightarrow H\) are linear bounded operators satisfying \(\|WT_{\Lambda}^{\ast}(I_{H}-Q)\|<1\).

Proof

First, we suppose that \(\|I_{H}-TT_{\Lambda}^{\ast}\|<1\) (\(T\in L(\bigoplus_{j\in J}V_{j}, H)\)). Then \(TT_{\Lambda}^{\ast}\) is invertible. Let \(W=T\) and \(Q=(TT_{\Lambda}^{\ast})^{-1}\). Then

$$\begin{aligned} S_{\Lambda}^{-1}T_{\Lambda}+W\bigl(I-T_{\Lambda}^{\ast}QS_{\Lambda}^{-1}T_{\Lambda}\bigr)&=S_{\Lambda}^{-1}T_{\Lambda}+T \bigl(I-T_{\Lambda}^{\ast}\bigl(TT_{\Lambda}^{\ast}\bigr)^{-1}S_{\Lambda}^{-1}T_{\Lambda}\bigr) \\ &=S_{\Lambda}^{-1}T_{\Lambda}+T-TT_{\Lambda}^{\ast}\bigl(TT_{\Lambda}^{\ast}\bigr)^{-1}S_{\Lambda}^{-1}T_{\Lambda}\\ &= S_{\Lambda}^{-1}T_{\Lambda}+T-S_{\Lambda}^{-1}T_{\Lambda}=T.\end{aligned} $$

Conversely, assume that \(T=S_{\Lambda}^{-1}T_{\Lambda}+W(I-T_{\Lambda}^{\ast}QS_{\Lambda}^{-1}T_{\Lambda})\). Then

$$\begin{aligned} TT_{\Lambda}^{\ast}&=\bigl(S_{\Lambda}^{-1}T_{\Lambda}+W \bigl(I-T_{\Lambda}^{\ast}QS_{\Lambda}^{-1}T_{\Lambda}\bigr)\bigr)T_{\Lambda}^{\ast}\\ &=S_{\Lambda}^{-1}T_{\Lambda}T_{\Lambda}^{\ast}+WT_{\Lambda}^{\ast}-WT_{\Lambda}^{\ast}QS_{\Lambda}^{-1}T_{\Lambda}T_{\Lambda}^{\ast}\\ &=I_{U}+WT_{\Lambda}^{\ast}-WT_{\Lambda}^{\ast}Q.\end{aligned} $$

Therefore

$$\bigl\Vert I_{H}-TT_{\Lambda}^{\ast}\bigr\Vert = \bigl\Vert WT_{\Lambda}^{\ast}(I_{H}-Q) \bigr\Vert < 1. $$

 □

Now, we turn to characterizing all approximately dual g-frames for a given g-frame.

Theorem 3.5

Let \(\{\Gamma_{j}\in L(H, V_{j})\}\) be a sequence, and let \(\{\Lambda _{j}\}_{j\in J}\) be a g-frame for H w.r.t. \(\{V_{j}\}_{j\in J}\) with the synthesis operator \(T_{\Lambda}\) and the frame operator \(S_{\Lambda}\). Then \(\{\Lambda_{j}\}_{j\in J}\) and \(\{\Gamma_{j}\}_{j\in J}\) are approximately dual g-frames if and only if

$$ \Gamma_{j}^{\ast}e_{j,k}=S_{\Lambda}^{-1} \Lambda_{j}^{\ast }e_{j,k}+W\tilde{e}_{j,k}- \sum_{j'\in J}\sum_{k'\in K_{j}}\bigl\langle QS_{\Lambda}^{-1}\Lambda _{j}^{\ast}e_{j,k}, \Lambda_{j'}^{\ast}e_{j',k'}\bigr\rangle W\tilde {e}_{j',k'},\quad \forall j\in J, k\in K_{j}, $$
(3.1)

where \(W: \bigoplus_{j\in J}V_{j}\rightarrow H\) and \(Q: H\rightarrow H\) are linear bounded operators satisfying \(\|WT_{\Lambda}^{\ast}(I_{H}-Q)\|<1\).

Proof

First, we assume that \(\{\Lambda_{j}\}_{j\in J}\) and \(\{ \Gamma_{j}\}_{j\in J}\) are approximately dual g-frames. By Lemmas 3.3 and 3.4 we have

$$ \Gamma_{j}^{\ast}e_{j,k}= \bigl(S_{\Lambda}^{-1}T_{\Lambda}+W\bigl(I-T_{\Lambda}^{\ast}QS_{\Lambda}^{-1}T_{\Lambda}\bigr)\bigr)\tilde {e}_{j,k}, $$
(3.2)

where I is the identity operator on \(\bigoplus_{j\in J}V_{j}\), and \(W: \bigoplus_{j\in J}V_{j}\rightarrow H\) and \(Q: H\rightarrow H\) are linear bounded operators satisfying \(\|WT_{\Lambda}^{\ast}(I_{U}-Q)\|<1\). Set \(z_{j,k}=W\tilde{e}_{j,k}\). We know that \(\{z_{j,k}\}_{j\in J, k\in K_{j}}\) is a Bessel sequence for H. Using the notations \(u_{j,k}:=\Lambda_{j}^{\ast}e_{j,k}\) and \(v_{j,k}:=\Gamma_{j}^{\ast }e_{j,k}\), we have

$$\bigl\{ \bigl\langle QS_{\Lambda}^{-1}u_{j,k}, u_{j',k'}\bigr\rangle \bigr\} _{j'\in J,k'\in K_{j}}\in l^{2} $$

for any \(j\in J\) and \(k\in K_{j}\). So \(\sum_{j'\in J}\sum_{k'\in K_{j}}\langle QS_{\Lambda}^{-1}u_{j,k}, u_{j',k'}\rangle z_{j',k'}\) converges unconditionally. By (3.2) we have

$$\begin{aligned} v_{j,k}&=S_{\Lambda}^{-1}T_{\Lambda}\tilde{e}_{j,k}+W\tilde {e}_{j,k}-WT_{\Lambda}^{\ast}QS_{\Lambda}^{-1}T_{\Lambda}\tilde{e}_{j,k} \\ &=S_{\Lambda}^{-1}u_{j,k}+z_{j,k}-WT_{\Lambda}^{\ast}QS_{\Lambda}^{-1}u_{j,k} \\ &=S_{\Lambda}^{-1}u_{j,k}+z_{j,k}-W \biggl(\sum _{j'\in J}\sum_{k'\in K_{j}}\bigl\langle \Lambda_{j'}QS_{\Lambda}^{-1}u_{j,k}, e_{j',k'}\bigr\rangle \tilde{e}_{j',k'} \biggr) \\ &= S_{\Lambda}^{-1}u_{j,k}+z_{j,k}-\sum _{j'\in J}\sum_{k'\in K_{j}}\bigl\langle QS_{\Lambda}^{-1}u_{j,k}, \Lambda_{j'}^{\ast}e_{j',k'} \bigr\rangle W\tilde {e}_{j',k'} \\ &=S_{\Lambda}^{-1}u_{j,k}+z_{j,k}-\sum _{j'\in J}\sum_{k'\in K_{j}}\bigl\langle QS_{\Lambda}^{-1}u_{j,k}, u_{j',k'}\bigr\rangle z_{j',k'}, \end{aligned}$$

that is,

$$\Gamma_{j}^{\ast}e_{j,k}=S_{\Lambda}^{-1} \Lambda_{j}^{\ast }e_{j,k}+W\tilde{e}_{j,k}- \sum_{j'\in J}\sum_{k'\in K_{j}}\bigl\langle QS_{\Lambda}^{-1}\Lambda _{j}^{\ast}e_{j,k}, \Lambda_{j'}^{\ast}e_{j',k'}\bigr\rangle W \tilde{e}_{j',k'} $$

for all \(j\in J\), \(k\in K_{j}\).

Now we prove the converse. Assume that (3.1) holds. For any \(f\in H\), using the notations \(u_{j,k}:=\Lambda_{j}^{\ast}e_{j,k}\), \(v_{j,k}:=\Gamma_{j}^{\ast}e_{j,k}\), and \(z_{j,k}:=W\tilde{e}_{j,k}\), by a standard argument we get that \(\sum_{j\in J}\sum_{k\in K_{j}}\langle f, u_{j,k}\rangle S^{-1}_{\Lambda}u_{j,k}\) converges unconditionally to f. Therefore

$$\begin{aligned} \sum_{j\in\mathcal{J}}\Gamma_{j}^{\ast} \Lambda_{j}f&= \sum_{j\in\mathcal{J}} \Gamma_{j}^{\ast} \sum_{k\in\mathcal{K}_{j}}\langle \Lambda_{j}f, e_{j,k}\rangle e_{j,k} \\ &=\sum_{j\in\mathcal{J}} \sum_{k\in\mathcal{K}_{j}} \bigl\langle f, \Lambda_{j}^{\ast }e_{j,k}\bigr\rangle \Gamma_{j}^{\ast}e_{j,k} \\ &= \sum_{j\in\mathcal{J}} \sum_{k\in\mathcal{K}_{j}} \langle f, u_{j,k}\rangle v_{j,k} \\ &=\sum_{j\in\mathcal{J}} \sum_{k\in\mathcal{K}_{j}} \langle f, u_{j,k}\rangle \biggl(S^{-1}_{\Lambda}u_{j,k}+z_{j,k}- \sum_{j^{\prime}\in\mathcal{J}} \sum_{k^{\prime}\in\mathcal{K}_{j}}\bigl\langle QS^{-1}_{\Lambda }u_{j,k}, u_{j^{\prime},k^{\prime}}\bigr\rangle z_{j^{\prime},k^{\prime }} \biggr) \\ &= \sum_{j\in\mathcal{J}} \sum_{k\in\mathcal{K}_{j}} \langle f, u_{j,k}\rangle S^{-1}_{\Lambda }u_{j,k}+ \sum_{j\in\mathcal{J}} \sum_{k\in\mathcal{K}_{j}} \langle f, u_{j,k}\rangle z_{j,k} \\ &\quad{} - \sum_{j\in\mathcal{J}} \sum_{k\in\mathcal{K}_{j}} \langle f, u_{j,k}\rangle\sum_{j^{\prime }\in\mathcal{J}} \sum _{k^{\prime}\in\mathcal{K}_{j}}\bigl\langle QS^{-1}_{\Lambda }u_{j,k}, u_{j^{\prime},k^{\prime}}\bigr\rangle z_{j^{\prime},k^{\prime }} \\ &= f+\sum_{j\in\mathcal{J}} \sum_{k\in\mathcal{K}_{j}} \langle f, u_{j,k}\rangle z_{j,k}-\sum _{j\in\mathcal{J}} \sum_{k\in\mathcal{K}_{j}} \biggl\langle Q \sum_{j^{\prime}\in \mathcal{J}} \sum_{k^{\prime}\in\mathcal{K}_{j}} \langle f, u_{j,k}\rangle S^{-1}_{\Lambda}u_{j,k}, u_{j^{\prime},k^{\prime}} \biggr\rangle z_{j^{\prime},k^{\prime}} \\ &=f+\sum_{j\in\mathcal{J}} \sum_{k\in\mathcal{K}_{j}} \langle f, u_{j,k}\rangle z_{j,k}- \sum _{j\in\mathcal{J}} \sum_{k\in\mathcal{K}_{j}}\langle Qf, u_{j^{\prime},k^{\prime }}\rangle z_{j^{\prime},k^{\prime}} \\ &= f+\sum_{j\in\mathcal{J}} \sum_{k\in\mathcal{K}_{j}} \langle f-Qf, u_{j,k}\rangle z_{j,k}\end{aligned} $$

for all \(f\in H\). Next, we prove that \(\{\Gamma_{j}\}_{j\in J}\) is a g-Bessel sequence for H w.r.t. \(\{V_{j}\}_{j\in J}\). Indeed,

$$\begin{aligned} \sum_{j\in J} \Vert \Gamma_{j}f \Vert ^{2}&=\sum_{j\in\mathcal{J}} \sum _{k\in K_{j}} \bigl\vert \langle\Gamma_{j}f, e_{j,k}\rangle \bigr\vert ^{2} \\ &=\sum_{j\in J} \sum_{k\in K_{j}} \bigl\vert \langle f, v_{j,k}\rangle \bigr\vert ^{2} \\ &= \sum_{j\in J} \sum_{k\in K_{j}} \biggl\vert \biggl\langle f, S^{-1}_{\Lambda }u_{j,k}+z_{j,k}- \sum_{j^{\prime}\in J} \sum_{k^{\prime}\in K_{j}}\bigl\langle QS^{-1}_{\Lambda}u_{j,k}, u_{j^{\prime},k^{\prime}}\bigr\rangle z_{j^{\prime},k^{\prime}} \biggr\rangle \biggr\vert ^{2} \\ &\leq C_{1} \biggl(\sum_{j\in\mathcal{J}} \sum _{k\in\mathcal{K}_{j}} \bigl\vert \bigl\langle f, S^{-1}_{\Lambda }u_{j,k}\bigr\rangle \bigr\vert ^{2}+\sum_{j\in J} \sum _{k\in K_{j}} \bigl\vert \langle f, z_{j,k}\rangle \bigr\vert ^{2} \\ &\quad{} +\sum_{j\in J} \sum_{k\in K_{j}} \biggl\vert \biggl\langle Q^{\ast}\sum_{j^{\prime}\in J} \sum_{k^{\prime}\in K_{j}}\langle f, z_{j^{\prime},k^{\prime }}\rangle u_{j^{\prime},k^{\prime}},S^{-1}_{\Lambda}u_{j,k}\biggr\rangle \biggr\vert ^{2} \biggr) \\ &\leq C_{2} \biggl( \Vert f \Vert ^{2}+ \biggl\Vert Q^{\ast}\sum_{j^{\prime}\in J} \sum _{k^{\prime}\in K_{j}}\langle f, z_{j^{\prime},k^{\prime }}\rangle u_{j^{\prime},k^{\prime}} \biggr\Vert ^{2} \biggr) \\ &\leq C_{3} \biggl( \Vert f \Vert ^{2}+\sum _{j^{\prime}\in J} \sum_{k^{\prime}\in K_{j}} \bigl\vert \langle f, z_{j^{\prime}k^{\prime }}\rangle \bigr\vert ^{2} \biggr) \\ &\leq C_{4} \Vert f \Vert ^{2} \end{aligned}$$

for all \(f\in H\), where \(C_{1}\), \(C_{2}\), \(C_{3}\), and \(C_{4}\) are different positive constants. Let \(T_{\Gamma}\) be the synthesis operator of \(\{\Gamma_{j}\}_{j\in J}\). Then

$$\begin{aligned} \bigl\Vert \bigl(I_{H}-T_{\Gamma}T^{\ast}_{\Lambda} \bigr)f \bigr\Vert &= \biggl\Vert \sum_{j\in J} \sum _{k\in K_{j}}\langle f-Qf, u_{j,k}\rangle z_{j,k} \biggr\Vert \\ &= \biggl\Vert \sum_{j\in J} \sum _{k\in K_{j}}\langle f-Qf, u_{j,k}\rangle W \tilde{e}_{j,k} \biggr\Vert \\ &= \biggl\Vert W\sum_{j\in J} \sum _{k\in K_{j}}\langle f-Qf, u_{j,k}\rangle \tilde{e}_{j,k} \biggr\Vert \\ &= \biggl\Vert W\sum_{j\in J} \sum _{k\in K_{j}}\bigl\langle \Lambda_{j}(f-Qf),e_{j,k} \bigr\rangle \tilde {e}_{j,k} \biggr\Vert \\ &= \bigl\Vert WT^{\ast}_{\Lambda}(f-Qf) \bigr\Vert \\ &\leq \bigl\Vert WT^{\ast}_{\Lambda}(I_{H}-Q) \bigr\Vert \Vert f \Vert \end{aligned}$$

for all \(f\in H\). Therefore \(\|I_{H}-T_{\Gamma}T^{\ast}_{\Lambda}\|<1\), and thus \(\{ \Lambda_{j}\}_{j\in J}\) and \(\{\Gamma_{j}\}_{j\in J}\) are approximately dual g-frames. □

4 Perturbations of approximately dual g-frames

The stability of frames is of great importance in frame theory, and it is studied widely by a lot of authors ([4, 18]). In this section, we show that, under some conditions, approximately dual g-frames and g-frames are stable under some perturbations. We first introduce some lemmas.

Lemma 4.1

([17])

Let \(\{\Lambda_{j}\}_{j\in J}\) be a g-frame for H w.r.t. \(\{V_{j}\} _{j\in J}\) with bounds A and B, \(\lambda_{1}, \lambda_{2}\in(-1, 1)\), \(\mu\geq0\), and \(\max\{\lambda_{1} + \frac{\mu}{\sqrt {A}},\lambda_{2}\}< 1\). If \(\{\Gamma_{j}\in L(H, V_{j})\}_{j\in J}\) satisfies

$$\biggl\Vert \sum_{j\in J_{1}}(\Lambda_{j}- \Gamma_{j})^{\ast}g_{j} \biggr\Vert \leq \lambda_{1} \biggl\Vert \sum_{j\in J_{1}} \Lambda_{j}^{\ast}g_{j} \biggr\Vert + \lambda_{2} \biggl\Vert \sum_{j\in J_{1}} \Gamma_{j}^{\ast}g_{j} \biggr\Vert +\mu \biggl(\sum _{j\in J_{1}} \Vert g_{j} \Vert ^{2} \biggr)^{\frac{1}{2}} $$

for an arbitrary finite subset \(J_{1}\subset J\) and \(g_{j}\in V_{j}\), then \(\{ \Gamma_{j}\}_{j\in J}\) is a g-frame for H w.r.t. \(\{V_{j}\}_{j\in J}\) with bounds

$$\frac{((1-\lambda_{1})\sqrt{A}-\mu)^{2}}{(1+\lambda_{2})^{2}},\qquad \frac{((1+\lambda_{1})\sqrt{B}+\mu)^{2}}{(1-\lambda_{2})^{2}}. $$

Lemma 4.2

([14])

Let \(\{\Lambda_{j}\}_{j\in J}\) be a g-frame for H w.r.t. \(\{V_{j}\} _{j\in J}\). Then for \(g_{j}\in V_{j}\) satisfying \(f=\sum_{j\in J}\Lambda _{j}^{\ast}g_{j}\), we have

$$\sum_{j\in J} \Vert g_{j} \Vert ^{2}\geq\sum_{j\in J} \Vert \tilde{ \Lambda}_{j}f \Vert ^{2}. $$

Lemma 4.3

([14])

\(\{\Lambda_{j}\}_{j\in J}\) is a g-Bessel sequence with an upper bound B if and only if

$$\biggl\Vert \sum_{j\in{J}_{1}}\Lambda_{j}^{\ast}g_{j} \biggr\Vert ^{2}\leq B\sum_{j\in{J}_{1}} \Vert g_{j} \Vert ^{2},\quad g_{j}\in V_{j}, $$

where \(J_{1}\) is an arbitrary finite subset of J.

Theorem 4.1

Let \(\Lambda_{j}\in L(H, V_{j})\), let \(\{\Gamma_{j}\}_{j\in J}\) be a g-frame for H w.r.t. \(\{V_{j}\}_{j\in J}\) with bounds A and B and the synthesis operator \(T_{\Lambda}\), and let \(\{\Delta_{j}\}_{j\in J}\) be alternate dual for \(\{\Gamma_{j}\}_{j\in J}\) with the upper bound C and the synthesis operator \(T_{\Delta}\). Assume that there are constants \(\lambda_{1}\), \(\mu\geq0\), and \(0\leq\lambda_{2}<1\) satisfying

$$ \biggl\Vert \sum_{j\in J_{1}}( \Gamma_{j}-\Lambda _{j})^{\ast}g_{j} \biggr\Vert \leq\lambda_{1} \biggl\Vert \sum _{j\in J_{1}}\Gamma _{j}^{\ast}g_{j} \biggr\Vert +\lambda_{2} \biggl\Vert \sum _{j\in J_{1}}\Lambda _{j}^{\ast}g_{j} \biggr\Vert +\mu \biggl(\sum_{j\in J_{1}} \Vert g_{j} \Vert ^{2} \biggr)^{\frac{1}{2}}, $$
(4.1)

where \(J_{1}\) is an arbitrary finite subset of J, and \(g_{j}\in V_{j}\). If

$$\lambda_{1}+\lambda_{2}\sqrt{BC} \biggl(1+\frac{\lambda_{1}+ \lambda_{2}+\frac{\mu}{\sqrt{B}}}{1-\lambda_{2}} \biggr)+\mu\sqrt{C}< 1, $$

then \(\{\Lambda_{j}\}_{j\in J}\) and \(\{\Delta_{j}\}_{j\in J}\) are approximately dual g-frames.

Proof

By Lemma 4.2 we have \(C\geq\frac{1}{A}\) and \(BC\geq\frac{B}{A}\geq1\). Note that

$$\lambda_{1}+\lambda_{2}\sqrt{BC} \biggl(1+\frac{\lambda_{1}+ \lambda_{2}+\frac{\mu}{\sqrt{B}}}{1-\lambda_{2}} \biggr)+\mu\sqrt{C}< 1. $$

It follows that \(\lambda_{1}+\frac{\mu}{\sqrt{A}}<1\). By Lemma 4.1 \(\{\Lambda_{j}\}_{j\in J}\) is a g-frame for H w.r.t. \(\{ V_{j}\}_{j\in J}\) with bounds

$$A\biggl(1-\frac{\lambda_{1}+\lambda_{2}+\frac{\mu}{\sqrt{A}}}{1+\lambda _{2}}\biggr)^{2},\qquad B\biggl(1+\frac{\lambda_{1}+\lambda_{2}+\frac{\mu}{\sqrt {B}}}{1-\lambda_{2}} \biggr)^{2}. $$

Denote by \(T_{\Lambda}\) the synthesis operator of \(\{\Lambda_{j}\} _{j\in J}\). From (4.1) we have

$$ \Vert T_{\Gamma}c-T_{\Lambda}c \Vert \leq \lambda_{1} \Vert T_{\Gamma}c \Vert +\lambda_{2} \Vert T_{\Lambda}c \Vert +\mu \Vert c \Vert _{\bigoplus_{j\in J}V_{j}} $$
(4.2)

for any \(c=\{c_{j}\}_{j\in J}\in\bigoplus_{j\in J}V_{j}\). Take \(c=T_{\Delta}^{\ast}f\) in (4.2). Then

$$\begin{aligned} \bigl\Vert \bigl(I_{H}-T_{\Lambda}T_{\Delta}^{\ast} \bigr)f \bigr\Vert &\leq \lambda_{1} \Vert f \Vert + \lambda_{2} \bigl\Vert T_{\Lambda}T_{\Delta}^{\ast}f \bigr\Vert +\mu \bigl\Vert T_{\Delta}^{\ast}f \bigr\Vert _{\bigoplus_{j\in J}V_{j}} \\ &\leq\lambda _{1} \Vert f \Vert +\lambda_{2}\sqrt{C} \Vert T_{\Lambda} \Vert \Vert f \Vert +\mu\sqrt{C} \Vert f \Vert \\ &\leq\lambda_{1} \Vert f \Vert +\lambda_{2}\sqrt{BC} \biggl(1+\frac{\lambda_{1}+\lambda_{2}+\frac{\mu}{\sqrt {B}}}{1-\lambda_{2}} \biggr) \Vert f \Vert +\mu\sqrt{C} \Vert f \Vert \\ &= \biggl(\lambda_{1}+\lambda_{2}\sqrt{BC} \biggl(1+ \frac{\lambda_{1}+\lambda_{2}+\frac{\mu}{\sqrt {B}}}{1-\lambda_{2}} \biggr) +\mu\sqrt{C} \biggr) \Vert f \Vert \end{aligned}$$

for any \(f\in H\). So

$$\bigl\Vert I_{H}-T_{\Lambda}T_{\Delta}^{\ast} \bigr\Vert \leq\lambda_{1}+\lambda _{2}\sqrt{BC} \biggl(1+ \frac{\lambda_{1}+\lambda_{2}+\frac{\mu}{\sqrt {B}}}{1-\lambda_{2}} \biggr) +\mu\sqrt{C}< 1. $$

Thus \(\{\Lambda_{j}\}_{j\in J}\) and \(\{\Delta_{j}\}_{j\in J}\) are approximately dual g-frames if \(\lambda_{1}+\lambda_{2}\sqrt{BC} (1+\frac{\lambda_{1}+ \lambda_{2}+\frac{\mu}{\sqrt{B}}}{1-\lambda_{2}} )+\mu\sqrt{C}<1\). □

From Theorem 4.1 we can obtain immediately the following corollary.

Corollary 4.1

Let \(\Lambda_{j}\in L(H, V_{j})\), let \(\{\Gamma_{j}\}_{j\in J}\) be a g-frame for H w.r.t. \(\{V_{j}\}_{j\in J}\) with bounds A and B and the synthesis operator \(T_{\Gamma}\), and let \(\{\Delta_{j}\}_{j\in J}\) be the canonical dual for \(\{\Gamma_{j}\}_{j\in J}\) with the synthesis operator \(T_{\Delta}\). Suppose that there are constants \(\lambda_{1}\), \(\mu\geq0\), and \(0\leq\lambda_{2}<1\) such that

$$ \biggl\Vert \sum_{j\in J_{1}}(\Gamma_{j}- \Lambda_{j})^{\ast }g_{j} \biggr\Vert \leq \lambda_{1} \biggl\Vert \sum_{j\in J_{1}} \Gamma_{j}^{\ast }g_{j} \biggr\Vert + \lambda_{2} \biggl\Vert \sum_{j\in J_{1}} \Lambda_{j}^{\ast }g_{j} \biggr\Vert +\mu \biggl( \sum_{j\in J_{1}} \Vert g_{j} \Vert ^{2} \biggr)^{\frac{1}{2}}, $$
(4.3)

where \(J_{1}\) is an arbitrary finite subset of J, and \(g_{j}\in V_{j}\). If \(\lambda_{1}+\lambda_{2}\sqrt{\frac{B}{A}} (1+\frac{\lambda_{1}+ \lambda_{2}+\frac{\mu}{\sqrt{B}}}{1-\lambda_{2}} )+\frac{\mu }{\sqrt{A}}<1\), then \(\{\Lambda_{j}\}_{j\in J}\) and \(\{\Delta_{j}\} _{j\in J}\) are approximately dual g-frames.

Note that \(\{\Gamma_{j}\}_{j\in J}\) is a Parseval g-frame for H w.r.t. \(\{V_{j}\}_{j\in J}\). Then \(\{\Gamma_{j}\}_{j\in J}\) is the canonical dual for itself. We have the following:

Corollary 4.2

Let \(\Lambda_{j}\in L(H, V_{j})\), and let \(\{\Gamma_{j}\}_{j\in J}\) be a Parseval g-frame for H w.r.t. \(\{V_{j}\}_{j\in J}\). Assume that there are constants \(\lambda,\mu\geq0\) such that

$$ \biggl\Vert \sum_{j\in J_{1}}(\Gamma_{j}- \Lambda_{j})^{\ast }g_{j} \biggr\Vert \leq\lambda \biggl\Vert \sum_{j\in J_{1}}\Gamma_{j}^{\ast }g_{j} \biggr\Vert +\mu \biggl(\sum_{j\in J_{1}} \Vert g_{j} \Vert ^{2} \biggr)^{\frac{1}{2}} $$
(4.4)

for an arbitrary finite subset \(J_{1}\subset J\) and \(g_{j}\in V_{j}\). If \(\lambda+\mu<1\), then \(\{\Lambda_{j}\}_{j\in J}\) and \(\{\Gamma_{j}\} _{j\in J}\) are approximately dual g-frames.

Corollary 4.3

Let \(\{\Lambda_{j}\in L(H, V_{j})\}_{j\in J}\) be a sequence, and let \(\{ \Gamma_{j}\}_{j\in J}\) be a g-frame for H w.r.t. \(\{V_{j}\}_{j\in J}\). Also, let \(\{\Delta_{j}\}_{j\in J}\) be an alternate dual for \(\{ \Gamma_{j}\}_{j\in J}\) with the upper bound C. If there exists a constant R such that \(CR<1\) and

$$ \sum_{j\in J} \bigl\Vert ( \Gamma_{j}-\Lambda_{j})f \bigr\Vert ^{2}\leq R \Vert f \Vert ^{2},\quad \forall f\in H, $$
(4.5)

then \(\{\Lambda_{j}\}_{j\in J}\) and \(\{\Delta_{j}\}_{j\in J}\) are approximately dual g-frames.

Proof

Take \(\lambda_{1}=\lambda_{2}=0\) and \(\mu=\sqrt{R}\) in Theorem 4.1. From Lemma 4.3 we know that (4.1) is equivalent to (4.5). Since \(CR<1\), we have that \(\{\Lambda_{j}\}_{j\in J}\) and \(\{\Delta _{j}\}_{j\in J}\) are approximately dual g-frames. □

Remark 4.1

Corollary 4.1 and Corollary 4.3 are Proposition 3.10(i) and Theorem 3.1(i) in [21], respectively. They are particular cases of our Theorem 4.1.

Theorem 4.2

Let \(\{\Lambda_{j}\}_{j\in J}\) be a g-frame for H w.r.t. \(\{V_{j}\}_{j\in J}\) with synthesis operator \(T_{\Lambda}\) and bounds A and B. Assume that \(\Gamma_{j}\in L(H, V_{j})\) for all \(j\in J\) and there exist constants \(\lambda_{1}, \lambda _{2}, \mu\geq0\) such that

$$ \biggl\Vert \sum_{j\in J_{1}}(\Lambda_{j}- \Gamma_{j})^{\ast }g_{j} \biggr\Vert \leq \lambda_{1} \biggl\Vert \sum_{j\in J_{1}} \Lambda_{j}^{\ast }g_{j} \biggr\Vert + \lambda_{2} \biggl\Vert \sum_{j\in J_{1}} \Gamma_{j}^{\ast }g_{j} \biggr\Vert +\mu \biggl(\sum _{j\in J_{1}} \Vert g_{j} \Vert ^{2} \biggr)^{\frac{1}{2}} $$
(4.6)

for an arbitrary finite subset \(J_{1}\subset J\) and \(g_{j}\in V_{j}\). If \(\lambda_{1}+\frac{\mu}{\sqrt{A}}<1\) and \(\lambda_{2}+(\lambda_{1}\sqrt {\frac{B}{A}}+\frac{\mu}{\sqrt{A}}) \frac{1+\lambda_{2}}{ 1-(\lambda_{1}+\frac{\mu}{\sqrt{A}})}<1\), then \(\{\Gamma_{j}\} _{j\in J}\) is a g-frame for H w.r.t. \(\{V_{j}\}_{j\in J}\), and \(\{ \tilde{\Gamma}_{j}\}_{j\in J}\) and \(\{\Lambda_{j}\}_{j\in J}\) are approximately dual g-frames.

Proof

We can prove the theorem by an argument similar to that of Theorem 4.1. □

5 Conclusions

For a given frame, it is usually not easy to find a dual frame. The notion of approximately dual frames was introduced by Christensen in 2010. It is a generalization of dual frames. In this paper, on one hand, we obtain the link between approximately dual g-frames and dual g-frames and characterize approximately dual g-frames. On the other hand, we give stability results of approximately dual g-frames, which cover the results obtained by other authors.