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Characterization and stability of approximately dual g-frames in Hilbert spaces

Journal of Inequalities and Applications20182018:192

https://doi.org/10.1186/s13660-018-1779-7

  • Received: 4 May 2018
  • Accepted: 17 July 2018
  • Published:

Abstract

This paper addresses approximately dual g-frames. First, we establish a connection between approximately dual g-frames and dual g-frames and obtain a characterization of approximately dual g-frames. Second, we give results on stability of approximately dual g-frames, which cover the results obtained by other authors.

Keywords

  • Frame
  • g-frame
  • Dual g-frame
  • Approximately dual g-frame

MSC

  • 42C15
  • 42C40

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 ([413]). 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 [1719]. 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.

Declarations

Acknowledgements

The authors would like to thank the reviewers for their suggestions, which greatly improved the readability of this paper.

Funding

The paper is supported by the National Natural Science Foundation of China (Grant No. 11271037).

Authors’ contributions

Both authors contributed to each part of this work equally and read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Information Engineering, Henan Finance University, Zhengzhou, P.R. China
(2)
School of Mathematics and Information Sciences, Henan University of Economics and Law, Zhengzhou, P.R. China

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