# Characterizing the R-duality of g-frames

## Abstract

In this paper, we define the g-Riesz-dual of a given special operator-valued sequence with respect to g-orthonormal bases for a separable Hilbert space. We demonstrate that the g-R-dual keeps some synchronous frame properties with the operator-valued sequence given. We also display some Schauder basis-like properties of the g-R-dual in the light of the properties of the given sequence. In particular, the g-R-dual can be characterized by the use of another sequence, related to the given sequence. Finally, a special sequence is constructed to build the relationship between an operator-valued sequence and a g-Riesz sequence.

## Introduction

Duality principles in Gabor theory play a fundamental role in analyzing the Gabor system. In [1], the authors described the concept of the Riesz-dual of a vector-valued sequence and illustrated the common frame properties for the give sequence and its R-dual. The conditions under which a Riesz sequence can be a R-dual of a given frame are investigated in [2]. In this paper, we are interested in the duality principles for g-frames. In [3], the g-R-dual was first defined, and some frame properties of g-R-dual were exhibited by the properties of the given operator-valued sequence. In this paper, our definition of g-R-dual in Sect. 2 is much weaker, and we characterize the g-R-dual with the analysis operator. The properties of the g-completeness, g-w-linearly independent, g-minimality of the g-R-dual is accounted in Sect. 3. In Sect. 4, we construct a sequence with a g-Riesz sequence and a given operator-valued sequence to consider the g-R-dual in a different way.

Throughout this paper, we use $$\mathbb{N}$$ to denote the set of all natural numbers, and assume that $$\{H_{i}\}_{i\in\mathbb{N}}$$ is a sequence of closed subspaces of a separable Hilbert space K, H is a separable Hilbert space. Denote by $$\{A_{i}\}_{i\in\mathbb{N}}$$, or for short $$\{A_{i}\}$$, a sequence of operators with $$A_{i}\in B(H,H_{i})$$ for any $$i\in\mathbb{N}$$. Suppose that $$B(H,H_{i})$$ denotes the collection of all the bounded linear operators from H into $$H_{i}$$, $$i\in\mathbb{N}$$. Denote by $$\bigoplus_{i\in \mathbb{N}}{H_{i}}$$ the orthogonal direct sum Hilbert space of $$\{ H_{i}\}_{i\in\mathbb{N}}$$, $$\{g_{i}\}:=\{g_{i}\}_{i\in\mathbb{N}}$$ for any $$\{g_{i}\}_{i\in\mathbb{N}}\in\bigoplus_{i\in\mathbb {N}}{H_{i}}$$.

In [10], Sun raised the concept of a g-frame as follows. Let $$A_{i}\in B(H,H_{i})$$, $$i\in\mathbb{N}$$. If there exist two constants $$a, b$$ such that

$$a \Vert f \Vert ^{2}\leq\sum_{i\in\mathbb {N}} \Vert A_{i}f \Vert ^{2}\leq b \Vert f \Vert ^{2},\quad \forall f\in H,$$

we call $$\{A_{i}\}$$ a g-frame for H. We call $$\{A_{i}\}$$ a tight g-frame for H if $$a=b$$. Specially, if $$a=b=1$$, $$\{A_{i}\}$$ is called a Parseval g-frame for H. If the inequalities above hold only for $$f\in \overline{\operatorname{span}} \{A^{*}_{i}H_{i}\} _{i\in\mathbb{N}}$$, we call $$\{A_{i}\}$$ a g-frame sequence for H. If only the right-hand inequality above holds, then we say that $$\{ A_{i}\}$$ is a g-Bessel sequence for H. If $$\overline{\operatorname{span}} \{A^{*}_{i}H_{i}\}_{i\in\mathbb{N}}=H$$, we say that $$\{A_{i}\}$$ is g-complete in H. If $$\{A_{i}\}$$ is g-complete and such that

$$a \bigl\Vert \{g_{i}\} \bigr\Vert ^{2}\leq\sum _{i\in\mathbb {N}} \bigl\Vert A_{i}^{*}g_{i} \bigr\Vert ^{2}\leq b \bigl\Vert \{g_{i}\} \bigr\Vert ^{2},\quad \forall \{g_{i}\} \in\bigoplus _{i\in\mathbb{N}}{H_{i}},$$

we call $$\{A_{i}\}$$ a g-Riesz basis for H. If the g-completeness is not satisfied, it is called a g-Riesz sequence for H. As we know, if $$\{A_{i}\}$$ is a g-frame for H, we define $$S_{A}f=\sum_{i\in\mathbb{N}}A^{*}_{i}A_{i}f$$ for any $$f\in H$$, then $$S_{A}$$ is a well-defined, bounded, positive, invertible operator by [10]. We call $$S_{A}$$ a frame operator of $$\{A_{i}\}$$. Another basic fact is that $$\{\widetilde{A}_{i}\}_{i\in\mathbb{N}}=\{ A_{i}S_{A}^{-1}\}_{i\in\mathbb{N}}$$ is a g-frame for H, we call it a canonical dual g-frame of $$\{A_{i}\}$$. Extensively, by [8], if a g-frame $$\{B_{i}\}$$ for H such that $$f=\sum_{i\in\mathbb {N}}B^{*}_{i}A_{i}f$$ for every $$f\in H$$, we say that it is a dual g-frame of $$\{A_{i}\}$$. Recently, g-frames in Hilbert spaces have been studied intensively; for more details see [4,5,6,7,8,9,10] and the references therein.

In the following we introduce some definitions and lemmas connected with the g-bases in Hilbert space which will be needed in the paper.

### Definition 1.1

([10])

If $$\{A_{i}\}$$ satisfies

1. (1)

$$\{A_{i}\}$$ is a g-orthonormal sequence for H, i.e., $$\langle A^{*}_{i}g_{i},A^{*}_{j}g_{j}\rangle=\delta _{ij}\langle g_{i},g_{j}\rangle$$ for any $$i, j\in\mathbb{N}$$, any $$g_{i}\in H_{i}, g_{j}\in H_{j}$$.

2. (2)

$$\{A_{i}\}$$ is g-complete in H.

We call $$\{A_{i}\}$$ a g-orthonormal basis for H. Obviously, (2) is equivalent to that $$\{A_{i}\}$$ is a Parseval g-frame for H by [5, Corollary 4.4], when (1) holds. Specially, if $$\{A_{i}\}$$ only satisfies $$A_{i}A^{*}_{j}=0$$ for any $$i, j\in\mathbb{N}$$, $$i\neq j$$, $$\{A_{i}\}$$ is called a g-orthogonal sequence for H.

The g-orthonormal basis is a special case that itself is g-biorthonormal. The following result shows that for the g-Riesz basis there also exists a g-biorthonormal sequence.

### Lemma 1.2

([10], Corollary 3.3)

Let $$\{A_{i}\}$$ be a g-Riesz basis for H. Then $$\{A_{i}\}$$ and $$\{ \widetilde{A}_{i}\}$$ are g-biorthonormal, where $$\{\widetilde{A}_{i}\}$$ is the canonical dual g-frame of $$\{A_{i}\}$$.

In this paper, we only interested in the case when the g-orthonormal basis for H exists, which is equivalent to the following result.

### Lemma 1.3

([5], Theorem 3.1)

Let H be a separable Hilbert space, $$\{H_{i}\}_{i\in\mathbb{N}}$$ be a sequence of separable Hilbert spaces. Then there exists a sequence $$\{ \varGamma_{i}\}$$, which is a g-orthonormal basis for H if and only if $$\operatorname{dim}H=\sum_{i\in\mathbb{N}}\operatorname{dim}H_{i}$$.

The concept of g-bases in Hilbert space is a generalization of the Schauder basis. Let $$\{A_{i}\}$$. If for any $$f\in H$$, there is a unique sequence $$\{g_{i}\}_{i\in\mathbb{N}}$$ with $$g_{i}\in H_{i}$$ for any $$i\in\mathbb{N}$$ such that $$f=\sum_{i\in\mathbb{N}}A_{i}^{*}g_{i}$$, we call $$\{A_{i}\}$$ a g-basis for H. If $$\{A_{i}\}$$ is a g-basis for $$\overline{\operatorname{span}} \{A_{i}^{*}H_{i}\}_{i\in\mathbb {N}}$$, $$\{A_{i}\}$$ is called a g-basic sequence for H. Moreover, If $$\sum_{i\in\mathbb{N}}A^{*}_{i}g_{i}=0$$ for $$\{g_{i}\} \in\bigoplus_{i\in\mathbb{N}}{H_{i}}$$, then $$g_{i}=0$$, we call $$\{A_{i}\}$$ g-w-linearly independent. If $$A^{*}_{j}g_{j}\notin \overline{\operatorname{span}}_{i\neq j} \{A^{*}_{i}g_{i}\}_{i\in \mathbb{N}}$$ for any $$\{g_{i}\}\in\bigoplus_{i\in\mathbb {N}}{H_{i}}$$ such that $$g_{i}\in H_{i}$$, $$g_{i}\neq0$$, any $$i\in \mathbb{N}$$, we call $$\{A_{i}\}$$ g-minimal. For more details as regards g-bases see [4].

## Duality for g-frame

Before giving the definition of g-R-dual, we introduce a lemma which is related to the g-Bessel sequence.

### Lemma 2.1

The sequence $$\{A_{i}\}$$ is a g-Bessel sequence for H if and only if $$\sum_{i\in\mathbb{N}}A^{*}_{i}g_{i}$$ is convergent for any $$\{g_{i}\}\in\bigoplus_{i\in\mathbb{N}}{H_{i}}$$, and is also equivalent to that $$\sum_{i\in\mathbb {N}} \Vert A_{i}f \Vert ^{2}<\infty$$ for every $$f\in H$$.

### Proof

Suppose $$\sum_{i\in\mathbb{N}}A^{*}_{i}g_{i}$$ is convergent for any $$\{g_{i}\}\in\bigoplus_{i\in\mathbb {N}}{H_{i}}$$. For any $$n\in\mathbb{N}$$, $$\{g_{i}\}\in \bigoplus_{i\in\mathbb{N}}{H_{i}}$$, we define $$T_{n}: \bigoplus_{i\in\mathbb{N}}{H_{i}}\rightarrow H, T_{n}\{g_{i}\}=\sum_{i=1}^{n}A^{*}_{i}g_{i}$$. Thus $$T_{n}$$ is bounded evidently. Since $$\{T_{n}\}_{n\in\mathbb{N}}$$ converges to T in the strong operator topology as $$n\rightarrow \infty$$, where $$T\{g_{i}\}=\sum_{i\in\mathbb {N}}A^{*}_{i}g_{i}$$ for every $$\{g_{i}\}\in\bigoplus_{i\in\mathbb{N}}{H_{i}}$$. Then T is bounded by the uniform boundedness principle in Banach space. The rest follows directly. □

For a g-Bessel sequence $$\{A_{i}\}$$, we can define the analysis operator as $$\theta_{A}: H\rightarrow \bigoplus_{i\in\mathbb{N}}{H_{i}}, \theta_{A}f=\{A_{i}f\}_{i\in\mathbb{N}}\text{ for any }f\in H$$, which is well defined and bounded obviously by Lemma 2.1.

### Definition 2.2

Let $$\{\varLambda_{i}\}$$, $$\{\varGamma_{i}\}$$ be two g-orthonormal bases for H. Suppose a sequence $$\{A_{i}\}$$ such that $$\sum_{i\in \mathbb{N}} \Vert A_{i}\varLambda^{*}_{j}g_{j} \Vert ^{2}<\infty$$ for any $$j\in \mathbb{N}$$, any $$g_{j}\in H_{j}$$. We define

$${\mathcal{A}}_{j}^{*}g_{j}=\sum _{i\in\mathbb {N}}\varGamma_{i}^{*}A_{i} \varLambda^{*}_{j}g_{j}, \quad\forall j\in\mathbb {N}, g_{j}\in H_{j}.$$

We call $$\{{\mathcal{A}}_{i}\}$$ a g-R-dual sequence of $$\{A_{i}\}$$.

### Remark 2.3

By [4, Theorem 4.4], for any $$j\in\mathbb{N}$$, $${\mathcal {A}}_{j}$$ is well defined if and only if $$\{A_{i}\varLambda^{*}_{j}g_{j}\} _{i\in\mathbb{N}}\in\bigoplus_{i\in\mathbb{N}} {H_{i}}$$ for any $$g_{j}\in H_{j}$$, i.e., $$\{A_{i}Q_{j}f\}_{i\in\mathbb{N}}\in \bigoplus_{i\in\mathbb{N}}{H_{i}}$$ for any $$f\in H$$, i.e., $$\{ A_{i}\}$$ is a g-Bessel sequence for $$\operatorname{ran}Q_{j}$$ by Lemma 2.1, where $$Q_{j}$$ is the orthogonal projection from H onto $$\overline{\operatorname{ran}} \varLambda_{j}^{*}$$. Obviously, $$\{A_{i}\}$$ may not be a g-Bessel sequence for H. The condition of our definition is weaker than that in [3, Definition 1.13]. Thus Definition 2.2 is equivalent to $${\mathcal{A}}_{j}=\sum_{i\in\mathbb {N}}\varLambda_{j}A_{i}^{*}\varGamma_{i}$$ for any $$j\in\mathbb{N}$$. By Definition 1.1, we get $$\varGamma_{k}{\mathcal {A}}^{*}_{j}=A_{k}\varLambda^{*}_{j}$$ for every $$i, k\in\mathbb{N}$$.

The following exhibits that the sequence $$\{A_{i}\}$$ satisfying Definition 2.2 shares the common properties with its g-R-dual $$\{{\mathcal{A}}_{i}\}$$. Similar results are referred to in [3, Theorem 2.2].

### Theorem 2.4

Let $$\{A_{i}\}$$ satisfy Definition 2.2, $$\{{\mathcal{A}}_{i}\}$$ be its g-R-dual defined in Definition 2.2. Then $$\{A_{i}\}$$ is a g-Bessel sequence for H if and only if $$\{{\mathcal{A}}_{i}\}$$ is a g-Bessel sequence for H. Moreover, they have the same upper bound.

### Proof

For every $$\{g_{i}\}\in\bigoplus_{i\in\mathbb{N}}{H_{i}}$$, let $$f=\sum_{i\in\mathbb{N}}\varLambda^{*}_{i}g_{i}$$, $$h=\sum_{i\in\mathbb {N}}\varGamma^{*}_{i}g_{i}$$. Suppose $$\{A_{i}\}$$ is a g-Bessel sequence for H and has an upper bound b. Since $$\theta_{\varLambda}, \theta_{\varGamma}: H\rightarrow \bigoplus_{i\in\mathbb{N}}{H_{i}}$$ are unitary,

\begin{aligned} \biggl\Vert \sum_{j\in\mathbb{N}}{\mathcal {A}}^{*}_{j}g_{j} \biggr\Vert ^{2}&= \biggl\Vert \sum_{j\in\mathbb{N}}\theta _{\varGamma}^{*} \theta_{\varGamma}{\mathcal{A}}^{*}_{j}g_{j} \biggr\Vert ^{2}= \biggl\Vert \sum_{j\in\mathbb{N}} \sum_{i\in\mathbb{N}}\varGamma ^{*}_{i} \varGamma_{i}{\mathcal{A}}^{*}_{j}g_{j} \biggr\Vert ^{2} \\ &= \biggl\Vert \sum_{j\in\mathbb{N}}\sum _{i\in\mathbb {N}}\varGamma^{*}_{i}A_{i} \varLambda^{*}_{j}g_{j} \biggr\Vert ^{2}= \biggl\Vert \sum_{i\in\mathbb{N}} \varGamma^{*}_{i}A_{i}f \biggr\Vert ^{2} \\ &= \bigl\Vert \theta_{\varGamma}^{*}\theta_{A}f \bigr\Vert ^{2}= \Vert \theta_{A}f \Vert ^{2}\leq b \Vert f \Vert ^{2} \\ &=b \bigl\Vert \theta_{\varGamma}^{*}\{g_{i}\} \bigr\Vert ^{2}=b \bigl\Vert \{g_{i}\} \bigr\Vert ^{2}. \end{aligned}

By Lemma 2.1, $$\{{\mathcal{A}}_{i}\}$$ is a g-Bessel sequence for H and has an upper bound b. The converse is similar. □

When $$\{A_{i}\}$$ is a g-Bessel sequence, there exists a unitary equivalence between $$\{\varLambda_{i}S_{A}^{\frac{1}{2}}\}$$ and the R-dual $$\{{\mathcal{A}}_{i}\}$$.

### Theorem 2.5

Let $$\{A_{i}\}$$ be a g-Bessel sequence for H, $$\{{\mathcal{A}}_{i}\}$$ be its g-R-dual defined in Definition 2.2. Then

1. (1)

$$\langle{\mathcal{A}}_{i}^{*}g_{i}, {\mathcal {A}}_{j}^{*}g_{k}\rangle=\langle S_{A}^{\frac{1}{2}}\varLambda _{j}^{*}g_{j}, S_{A}^{\frac{1}{2}}\varLambda_{i}^{*}g_{i}\rangle$$ for any $$i, j\in\mathbb{N}$$, any $$g_{i}\in H_{i}, g_{j}\in H_{j}$$.

2. (2)

$$\Vert \sum_{i\in\mathbb{N}}{\mathcal {A}}_{i}^{*}g_{i} \Vert = \Vert \sum_{i\in\mathbb{N}}S_{A}^{\frac {1}{2}}\varLambda_{i}^{*}g_{i} \Vert$$ for any $$\{g_{i}\}\in\bigoplus_{i\in\mathbb{N}} {H_{i}}$$.

3. (3)

there exists an isometric operator T from $$\overline{\operatorname{ran}} S_{A}^{\frac{1}{2}}\theta_{\varLambda}^{*}$$ onto $$\overline{\operatorname{ran}} \theta_{\mathcal{A}}^{*}$$ such that $${\mathcal {A}}_{i}T=\varLambda_{i}S_{A}^{\frac{1}{2}}$$ for any $$i\in \mathbb{N}$$.

### Proof

(1) Since $$\{A_{i}\}$$ is a g-Bessel sequence for H, so is $$\{{\mathcal {A}}_{i}\}$$ by Theorem 2.4. Then, for any $$i, j\in\mathbb {N}$$, any $$g_{i}\in H_{i}, g_{j}\in H_{j}$$, we have

\begin{aligned} \bigl\langle {\mathcal{A}}_{i}^{*}g_{i}, { \mathcal{A}}_{j}^{*}g_{k} \bigr\rangle &= \bigl\langle \theta_{\mathcal{A}}^{*}\{\delta_{ik}g_{i} \}_{k}, \theta _{\mathcal{A}}^{*}\{\delta_{jk}g_{j} \}_{k} \bigr\rangle \\ &= \bigl\langle \theta_{\varGamma}^{*}\theta_{A} \theta_{\varLambda}^{*}\{ \delta_{ik}g_{i} \}_{k}, \theta_{\varGamma}^{*}\theta_{A}\theta _{\varLambda}^{*}\{\delta_{jk}g_{j} \}_{k} \bigr\rangle \\ &= \bigl\langle S_{A}^{\frac{1}{2}}\varLambda_{i}^{*}g_{i}, S_{A}^{\frac {1}{2}}\varLambda_{j}^{*}g_{j} \bigr\rangle . \end{aligned}

(2) It is direct by (1).

(3) Define $$T^{*}:\operatorname{ran}\theta_{\mathcal{A}}^{*}\rightarrow \operatorname{ran}S_{A}^{\frac{1}{2}}$$, $$T^{*}(\sum_{i\in\mathbb {N}}{\mathcal{A}}^{*}_{i}g_{i})=\sum_{i\in\mathbb{N}}S_{A}^{\frac {1}{2}}\varLambda_{i}^{*}g_{i}$$ for any $$\{g_{i}\}\in\bigoplus_{i\in\mathbb{N}} {H_{i}}$$. It is easy to verify $$T^{*}$$ is well defined by (2). We can extend T to an isometric operator from $$\overline{\operatorname {ran}} S_{A}^{\frac{1}{2}}\theta_{\varLambda}^{*}$$ onto $$\overline{\operatorname {ran}} \theta_{\mathcal{A}}^{*}$$. We still denote the operator as T for convenience. □

In the following results we show the properties of g-R-dual in the case that $$\{A_{i}\}$$ is a g-frame sequence by the corresponding analysis operators. The results are similar to the conclusions in [3, Corollary 2.6].

### Theorem 2.6

Let $$\{A_{i}\}$$ satisfy Definition 2.2, $$\{{\mathcal{A}}_{i}\}$$ be its g-R-dual defined in Definition 2.2. Then $$\{A_{i}\}$$ is a g-frame sequence for H if and only if $$\{{\mathcal{A}}_{i}\}$$ is a g-frame sequence for H with the same frame bounds. Specially, in this case the following are equivalent:

1. (1)

$$\{A_{i}\}$$ is a g-frame for H with the frame bounds $$a, b$$.

2. (2)

$$\{{\mathcal{A}}_{i}\}$$ is a g-Riesz sequence for H with the frame bounds $$a, b$$.

3. (3)

There exists $$0< b_{1}<\infty$$ such that $$\sum_{i\in\mathbb{N}} \Vert A_{i}Pf \Vert ^{2}\leq b_{1}\sum_{i\in \mathbb{N}} \Vert A_{i}f \Vert ^{2}$$ for any $$f\in H$$, where P is an arbitrary orthogonal projection on H.

4. (4)

There exists $$0< b_{1}<\infty$$ such that $$\sum_{i\in\mathbb{N}} \Vert A_{i}P_{n}f \Vert ^{2}\leq b_{1}\sum_{i\in\mathbb{N}} \Vert A_{i}f \Vert ^{2}$$ for any $$f\in H$$, where $$P_{n}$$ is the orthogonal projection from H onto $$\overline{\operatorname{span}} \{ \varLambda_{i}^{*}H_{i}\}_{i=1}^{n}$$ for any $$n\in\mathbb{N}$$.

### Proof

The case of the g-Bessel upper bound we get easily by Theorem 2.4. We now show the case of the lower bound in a similar way as the proof of Theorem 2.4.

Because $$\{A_{i}\}$$, $$\{{\mathcal{A}}_{i}\}$$ are g-Bessel sequences, we easily have $$\theta_{A}=\theta_{\varGamma}\theta_{\mathcal {A}}^{*}\theta_{\varLambda}$$. Then $$g\in\operatorname{ker}\theta_{A}$$ if and only if $$g\in\operatorname{ker}\theta_{\mathcal{A}}^{*}\theta _{\varLambda}$$, i.e., $$\theta_{\varLambda}g\in\operatorname{ker}\theta _{\mathcal{A}}^{*}$$. Hence, $$g\in(\operatorname{ker}\theta_{A})^{\bot}$$ if and only if $$\theta_{\varLambda}g\in(\operatorname{ker}\theta_{\mathcal {A}}^{*})^{\bot}$$ since $$\theta_{\varLambda}$$ is unitary.

Evidently, $$\{A_{i}\}$$ is a g-frame sequence for H if and only if for any $$f\in\operatorname{ran}\theta^{*}_{A}$$, one has $$a \Vert f \Vert ^{2}\leq\sum_{i\in\mathbb{N}} \Vert A_{i}f \Vert ^{2}= \Vert \theta _{A}f \Vert ^{2}\leq b \Vert f \Vert ^{2}$$, i.e.,

$$a \Vert \theta_{\varLambda}f \Vert ^{2}= \bigl\Vert \theta_{\mathcal {A}}^{*}\theta_{\varLambda}f \bigr\Vert ^{2}\leq b \Vert f \Vert ^{2}=b \Vert \theta_{\varLambda}f \Vert ^{2},$$

which is equivalent to $$\{{\mathcal{A}}_{i}\}$$ is a g-frame sequence for H.

The equivalence of (1) and (2) is obvious since $$(\operatorname{ker}\theta _{A})^{\bot}=\{0\}$$ if and only if $$(\operatorname{ker}\theta_{\mathcal {A}}^{*})^{\bot}=\{0\}$$ by the proof above.

(1) (3). Let $$\{A_{i}\}$$ be a g-frame for H with the frame bounds $$a, b$$. Take P as an arbitrary orthogonal projection on H. For any $$f=f_{1}+f_{2}\in H$$, where $$f_{1}\in\operatorname{ran}P, f_{2}\in\operatorname{ker}P$$, we have

$$\sum_{i\in\mathbb{N}} \Vert A_{i}Pf \Vert ^{2}=\sum_{i\in\mathbb{N}} \Vert A_{i}f_{1} \Vert ^{2}\leq b \Vert f \Vert ^{2}\leq a^{-1}b\sum_{i\in\mathbb{N}} \Vert A_{i}f \Vert ^{2}.$$

(3) (4) is direct.

(4) (2). It is obvious by Theorem 3.3. □

The following result was given in [3, Theorem 4.1], we here give a simple illustration by the use of the analysis operators.

### Lemma 2.7

Let $$\{A_{i}\}, \{B_{i}\}$$ be two g-frames for H, $$\{{\mathcal{A}}_{i}\}$$, $$\{{\mathcal{B}}_{i}\}$$ be their g-R-dual sequences defined in Definition 2.2, respectively. Then $$\{A_{i}\}$$ is a dual g-frame of $$\{B_{i}\}$$ if and only if $$\langle{\mathcal{A}}^{*}_{i}g_{i}, {\mathcal{B}}^{*}_{j}g_{j}\rangle =\delta_{ij}\langle g_{i},g_{j}\rangle$$ for any $$i, j\in\mathbb{N}$$, any $$g_{i}\in H_{i}$$, $$g_{j}\in H_{j}$$.

### Proof

By Definition 2.2, we get $$\theta_{\mathcal{A}}=\theta _{\varLambda}\theta^{*}_{A}\theta_{\varGamma}$$, $$\theta_{\mathcal {B}}=\theta_{\varLambda}\theta^{*}_{B}\theta_{\varGamma}$$. Then $$\theta_{\mathcal{A}}\theta_{\mathcal{B}}^{*}=\theta_{\varLambda }\theta^{*}_{A}\theta_{B}\theta_{\varLambda}^{*}$$. Obviously, $$\theta^{*}_{A}\theta_{B}=I$$ if and only if $$\theta _{\mathcal{A}}\theta_{\mathcal{B}}^{*}=I_{\bigoplus_{i\in \mathbb{N}}{H_{i}}}$$, i.e., $$\langle{\mathcal{A}}^{*}_{i}g_{i}, {\mathcal{B}}^{*}_{j}g_{j}\rangle=\delta_{ij}\langle g_{i},g_{j}\rangle$$ for any $$i, j\in\mathbb{N}$$, any $$g_{i}\in H_{i}$$, $$g_{j}\in H_{j}$$. □

The following shows that the g-R-dual of the canonical dual g-frame is the “minimal” and has the “smallest distance” with $$\{A_{i}\}$$ among the g-R-duals of all the alternate dual g-frames, which is a generalization of the result in [3, Theorem 4.5].

### Theorem 2.8

Let $$\{A_{i}\}$$ be a g-frame for H, $$\{\widetilde{A}_{i}\}$$ be the canonical dual g-frame of $$\{A_{i}\}$$, $$\{B_{i}\}$$ be a dual g-frame of $$\{A_{i}\}$$. $$\{{\mathcal{A}}_{i}\}$$ and $$\{{\mathcal{B}}_{i}\}$$ are the corresponding g-R-duals defined in Definition 2.2, respectively. Then the following are equivalent:

1. (1)

$$B_{i}=\widetilde{A}_{i}$$ for every $$i\in\mathbb{N}$$.

2. (2)

$$\Vert {\mathcal{B}}^{*}g_{i} \Vert \leq \Vert {\mathcal {C}}_{i}^{*}g_{i} \Vert$$ for every $$i\in\mathbb{N}$$, $$g_{i}\in H_{i}$$, where $$\{C_{i}\}$$ is an arbitrary dual g-frame of $$\{A_{i}\}$$, $$\{ \mathcal{C}_{i}\}$$ is the g-R-dual of $$\{C_{i}\}$$.

3. (3)

$$\Vert {\mathcal{B}}_{i}^{*}g_{i}-{\mathcal {A}}_{i}^{*}g_{i} \Vert \leq \Vert {\mathcal{C}}_{i}^{*}g_{i}-{\mathcal {A}}_{i}^{*}g_{i} \Vert$$ for every $$i\in\mathbb{N}$$, $$g_{i}\in H_{i}$$, where $$\{C_{i}\}$$ is an arbitrary dual g-frame of $$\{A_{i}\}$$, $$\{ \mathcal{C}_{i}\}$$ is the g-R-dual of $$\{C_{i}\}$$.

### Proof

(1) (2). By [3, Theorem 4.4], we obtain $${\mathcal{B}}_{i}=\widetilde{{\mathcal{A}}}_{i}+\Delta_{i}$$ for any $$i\in\mathbb{N}$$, where $$\{\Delta_{i}\}$$ is a g-Bessel sequence for H such that $$\operatorname{ran}\theta_{\Delta}^{*}\subset(\operatorname {ran}\theta_{\mathcal{A}}^{*})^{\bot}$$. Then, for every $$\{g_{i}\}\in \bigoplus_{i\in\mathbb{N}} {H_{i}}$$, we get

$$\bigl\Vert \theta_{\mathcal{B}}^{*}\{g_{i}\} \bigr\Vert ^{2}= \bigl\Vert \theta _{\widetilde{{\mathcal{A}}}}^{*} \{g_{i}\}+\theta_{\Delta}^{*}\{g_{i}\} \bigr\Vert ^{2}\geq \bigl\Vert \theta_{\widetilde{{\mathcal{A}}}}^{*} \{g_{i}\} \bigr\Vert ^{2}.$$

Specially, if we take $$\{\delta_{ij}g_{i}\}_{j\in\mathbb{N}}$$, then $$\Vert {\mathcal{B}}_{i}^{*}g_{i} \Vert \geq \Vert {\widetilde{{\mathcal {A}}}}_{i}^{*}g_{i} \Vert$$. Hence, $$B_{i}=\widetilde{A}_{i}$$ if and only if $$\Delta_{i}=0$$ for any $$i\in\mathbb{N}$$.

(2) (3). By Lemma 2.7, for any $$i\in \mathbb{N}$$, we obtain

$$\bigl\Vert {\mathcal{B}}_{i}^{*}g_{i}-{\mathcal {A}}_{i}^{*}g_{i} \bigr\Vert ^{2}= \bigl\Vert {\mathcal{B}}_{i}^{*}g_{i} \bigr\Vert ^{2}+ \bigl\Vert {\mathcal {A}}_{i}^{*}g_{i} \bigr\Vert ^{2}-2.$$

Similarly, $$\Vert {\widetilde{{\mathcal{A}}}}_{i}^{*}g_{i}-{\mathcal {A}}_{i}^{*}g_{i} \Vert = \Vert {\widetilde{{\mathcal {A}}}}_{i}^{*}g_{i} \Vert ^{2}+ \Vert {\mathcal{A}}_{i}^{*}g_{i} \Vert ^{2}-2$$. Thus the equivalence is direct. □

## Characterization of the Schauder basis-like properties of g-R-dual

Suppose $$\{A_{i}\}$$ is a g-Bessel sequence for H, $$\{{\mathcal {A}}_{i}\}$$ is its g-R-dual defined in Definition 2.2. We will characterize the Schauder basis-like properties (g-completeness, g-w-linearly independence, g-minimality) of $$\{{\mathcal{A}}_{i}\}$$ in terms of $$\{A_{i}\}$$.

### Theorem 3.1

Let $$\{A_{i}\}$$ be a g-Bessel sequence for H, $$\{{\mathcal{A}}_{i}\}$$ be its g-R-dual defined in Definition 2.2. Then the following are equivalent:

1. (1)

$$\{A_{i}\}$$ is g-complete.

2. (2)

$$\{{\mathcal{A}}_{i}\}$$ is g-w-linearly independent.

3. (3)

If $$\lim_{n\rightarrow\infty} \Vert \theta _{A}x_{n} \Vert ^{2}=0$$, then $$\{g_{i}\}=0$$, where $$x_{n}=\sum_{i=1}^{n}\varLambda_{i}^{*}g_{i}\in H$$ for any $$n\in\mathbb{N}$$ and any $$\{g_{i}\}\in\bigoplus_{i\in\mathbb{N}} {H_{i}}$$.

### Proof

(1) (2). By Definition 2.2, $$\theta _{\mathcal{A}}^{*}=\theta_{\varGamma}^{*}\theta_{A}\theta_{\varLambda }^{*}$$. For arbitrary $$\{g_{i}\}\in\bigoplus_{i\in\mathbb{N}} {H_{i}}$$, we have $$\{g_{i}\}\in\operatorname{ker}\theta_{\mathcal {A}}^{*}$$ if and only if $$\theta_{\varLambda}^{*}\{g_{i}\}\in\operatorname {ker}\theta_{A}$$. Then $$\{A_{i}\}$$ is g-complete if and only if $$\operatorname{ker}\theta_{\mathcal{A}}^{*}=\{0\}$$, i.e., $$\{{\mathcal {A}}_{i}\}$$ is g-w-linearly independent.

(2) (3). It is evident as $$\Vert \theta _{A}x_{n} \Vert ^{2}= \Vert \theta_{\mathcal{A}}^{*}\theta_{\varLambda}x_{n} \Vert ^{2}$$. □

Now we have the next special result. By [4, Theorem 5.2], if $$\{A_{i}\}$$ is a g-frame sequence for H, the existing condition of the g-biorthonormal sequence means the minimality of $$\{A_{i}\}$$.

### Theorem 3.2

Let $$\{A_{i}\}$$ be a g-Bessel sequence for H, $$\{{\mathcal{A}}_{i}\}$$ defined in Definition 2.2 be its g-R-dual. If there exists a sequence $$\{\Delta_{i}\}$$ which is g-biorthonormal with $$\{{\mathcal {A}}_{i}\}$$ such that $$\Delta_{i}^{*}$$ is injective for any $$i\in \mathbb{N}$$, then

1. (1)

there are constants $$0< c_{i}\leq1$$ for arbitrary $$i\in\mathbb{N}$$ such that $$\Vert c_{i}g_{i} \Vert \leq \Vert \sum_{j\in \mathbb{N}}{\mathcal{A}}_{j}^{*}g_{j} \Vert$$ for any $$\{g_{i}\}\in\bigoplus_{i\in\mathbb{N}} {H_{i}}$$;

2. (2)

there are constants $$0< a_{i}$$ for arbitrary $$i\in \mathbb{N}$$ such that

$$\bigl\Vert \{a_{i}g_{i}\}_{i\in\mathbb{N}} \bigr\Vert ^{2}\leq\sum_{j\in\mathbb{N}} \bigl\Vert A_{j}\theta_{\varLambda}^{*}\{g_{i}\} \bigr\Vert ^{2},\quad \forall \{g_{i}\}\in\bigoplus _{i\in\mathbb {N}} {H_{i}}.$$

Moreover, (1) and (2) are equivalent.

### Proof

Take arbitrary $$h_{i}\in H_{i}$$ and $$\Vert h_{i} \Vert =1$$ and let $$c_{i}=\min\{ 1, \frac{1}{ \Vert \Delta_{i} \Vert }\}$$ for every $$i\in\mathbb{N}$$. Since $$\langle{\mathcal{A}}_{i}^{*}g_{i}, \Delta_{j}^{*}g_{j}\rangle =\delta_{ij}\langle g_{i}, g_{j}\rangle$$ for any $$i, j\in\mathbb {N}$$, $$g_{i}\in H_{i}$$ $$g_{j}\in H_{j}$$, we have

\begin{aligned} \biggl\Vert \sum_{j\in\mathbb{N}}{\mathcal{A}}_{j}^{*}g_{j} \biggr\Vert &=\sup_{ \Vert f \Vert =1,f\in H} \biggl\vert \biggl\langle \sum _{j\in\mathbb {N}}{\mathcal{A}}_{j}^{*}g_{j},f \biggr\rangle \biggr\vert \\ &\geq \biggl\vert \biggl\langle \sum_{j\in\mathbb{N}}{\mathcal {A}}_{j}^{*}g_{j},\frac{1}{ \Vert \Delta_{i}^{*}h_{i} \Vert }\Delta _{i}^{*}h_{i} \biggr\rangle \biggr\vert \\ &\geq \biggl\vert \biggl\langle \sum_{j\in\mathbb{N}}{\mathcal {A}}_{j}^{*}g_{j},\frac{1}{ \Vert \Delta_{i} \Vert } \Delta_{i}^{*}h_{i} \biggr\rangle \biggr\vert \\ &\geq\vert c_{i}\vert\biggl| \biggl\langle \sum _{j\in\mathbb{N}}{\mathcal {A}}_{j}^{*}g_{j}, \Delta_{i}^{*}h_{i} \biggr\rangle \biggr\vert = | c_{i}|\bigl| \langle g_{i},h_{i}\rangle \bigr|. \end{aligned}

By the arbitrariness of $$h_{i}$$, we have $$|c_{i} \Vert g_{i} \Vert \leq \Vert \sum_{j\in\mathbb{N}}{\mathcal{A}}_{j}^{*}g_{j} \Vert$$.

Take $$a_{i}=\frac{c_{i}}{2^{i}}$$ for every $$i\in\mathbb{N}$$. For any $$\{g_{i}\}\in\bigoplus_{i\in\mathbb{N}} {H_{i}}$$, we obtain

\begin{aligned} \bigl\Vert \{a_{i}g_{i}\} \bigr\Vert ^{2}&= \sum_{i\in\mathbb{N}} \biggl\Vert \frac {c_{i}}{2^{i}}g_{i} \biggr\Vert ^{2}=\sum_{i\in\mathbb{N}} \frac {1}{2^{2i}} \Vert c_{i}g_{i} \Vert ^{2} \\ &\leq\sum_{i\in\mathbb{N}}\frac{1}{2^{2i}}\sup _{i\in\mathbb{N}} \Vert c_{i}g_{i} \Vert ^{2} \\ &\leq \biggl\Vert \sum_{j\in\mathbb{N}}{\mathcal {A}}_{j}^{*}g_{j} \biggr\Vert =\sum _{j\in\mathbb{N}} \bigl\Vert A_{j}\theta _{\varLambda}^{*} \{g_{i}\} \bigr\Vert ^{2}. \end{aligned}

The converse is evident since $$\Vert a_{i}g_{i} \Vert ^{2}\leq \Vert \{a_{i}g_{i}\} \Vert ^{2}$$. □

In the following we illustrate that the g-R-dual $$\{{\mathcal{A}}_{i}\}$$ is a g-basic sequence by the properties of $$\{A_{i}\}$$, which also shows the conclusion of Theorem 2.6 from another perspective. It can be realized as a kind of g-completeness of $$\{{\mathcal{A}}_{i}\}$$.

### Theorem 3.3

Let $$\{A_{i}\}$$ be a g-frame sequence for H, $$\{{\mathcal{A}}_{i}\}$$ defined in Definition 2.2 be its g-R-dual. Let $$P_{n}$$ be the orthogonal projection from H onto $$N_{n}:=\overline{\operatorname{span}} \{ \varLambda^{*}_{i}H_{i}\}_{i=1}^{n}$$ for any $$n\in\mathbb{N}$$. Then the following are equivalent:

1. (1)

$$\{{\mathcal{A}_{i}}\}$$ a g-basic sequence for H.

2. (2)

There exists a constant $$0< b<\infty$$ such that $$\sum_{i\in\mathbb{N}} \Vert A_{i}P_{n}f \Vert ^{2}\leq b\sum_{i\in\mathbb{N}} \Vert A_{i}f \Vert ^{2}$$ for any $$n\in\mathbb{N}$$, any $$f\in H$$.

3. (3)

There exists a constant $$0< b<\infty$$ such that $$S_{AP_{n}}\leq bS_{A}$$ for any $$n\in\mathbb{N}$$, where $$S_{AP_{n}}$$ is the frame operator of the g-Bessel sequence $$\{A_{i}P_{n}\}_{i\in \mathbb{N}}$$.

In this case, we have

$$\operatorname{ran}\theta^{*}_{A}=\overline{ \operatorname{span}} \biggl\{ \varLambda_{i}^{*}g_{i}: \sum_{i\in\mathbb{N}} \bigl\Vert A_{i}\varLambda _{i}^{*}g_{i} \bigr\Vert ^{2}\neq0, \forall i\in\mathbb{N}, g_{i}\in H_{i} \biggr\} .$$

### Proof

Let $${\mathbb{I}}=\{j\in{\mathbb{N}}:{\mathcal{A}}_{j}^{*}=\theta _{\varGamma}^{*}\theta_{A}\varLambda^{*}_{j}\neq0\}$$. Without loss of generality, we can suppose $${\mathcal{A}}_{i}\neq0$$ for any $$i\in \mathbb{N}$$.

(1) (2). By [4, Theorem 3.3], $$\{{\mathcal {A}_{i}}\}$$ is a g-basic sequence for H if and only if there exists a constant $$0< b<\infty$$ such that, for arbitrary $$n\leq m$$, any $$\{ g_{i}\}\in\bigoplus_{i\in\mathbb{N}} {H_{i}}$$, one has

$$\Biggl\Vert \sum_{i=1}^{n} {\mathcal {A}}_{i}^{*}g_{i} \Biggr\Vert ^{2}\leq b \Biggl\Vert \sum_{i=1}^{m}{\mathcal {A}}_{i}^{*}g_{i} \Biggr\Vert ^{2}=b \sum_{i\in\mathbb{N}} \Vert A_{i}x \Vert ^{2},$$

where $$x=\sum_{i=1}^{m}\varLambda_{i}^{*}g_{i}$$. Since $$P_{n}\varLambda _{i}^{*}=0$$ for every $$i\in\mathbb{N}$$ such that $$n< i\leq m$$, $$\sum_{i=1}^{n}\varLambda_{i}^{*}g_{i}=P_{n}x$$. Similarly, we have $$\Vert \sum_{i=1}^{n} {\mathcal{A}}_{i}^{*}g_{i} \Vert ^{2}=\sum_{i\in\mathbb {N}} \Vert A_{i}P_{n}x \Vert ^{2}$$.

(2) (3). (2) is equivalent to $$\langle S_{AP_{n}}f, f\rangle=\langle\theta_{A}P_{n} f, \theta _{A}P_{n} f\rangle\leq b\langle Sf,f\rangle$$ for any $$f\in H$$, which is obvious.

By [4, Lemma 2.16], $$\{{\mathcal{A}}_{i}\}$$ is a g-Riesz sequence for H. Then $${\mathcal{A}}_{i}\neq0$$ for any $$i\in\mathbb {N}$$. By Definition 2.2, we have $${\mathcal{A}}_{i}^{*}=\theta ^{*}_{\varGamma}\theta_{A}\varLambda^{*}_{i}$$. Then $$\theta_{A}\varLambda ^{*}_{i}\neq0$$, i.e., $$\sum_{i\in\mathbb{N}} \Vert A_{i}\varLambda _{i}^{*}g_{i} \Vert ^{2}\neq0$$ for any $$i\in\mathbb{N}$$, $$g_{i}\in H_{i}$$. Hence,

$$\overline{\operatorname{span}} \biggl\{ \varLambda_{i}^{*}g_{i}: \sum_{i\in\mathbb{N}} \bigl\Vert A_{i} \varLambda_{i}^{*}g_{i} \bigr\Vert ^{2} \neq0, \forall i\in{\mathbb{N}}, g_{i}\in H_{i} \biggr\} =H.$$

Therefore, we only need to show the g-completeness of $$\{A_{i}\}$$ in H.

Suppose there exists $$f\in H$$, $$f\neq0$$ such that $$\langle A_{i}^{*}g_{i}, f\rangle=0$$ for arbitrary $$i\in\mathbb{N}$$, $$g_{i}\in H_{i}$$. Obviously, there is a sequence $$\{f_{i}\}\in\bigoplus_{i\in\mathbb{N}} {H_{i}}$$ such that $$f=\sum_{i\in\mathbb {N}}\varLambda_{i}^{*}f_{i}$$. Assume $$k\in{\mathbb{N}}$$ is the smallest positive integer such that $$f_{i}\neq0$$. Then $$P_{k}f=\varLambda _{k}^{*}f_{k}$$. We get

$$0\neq\sum_{i\in\mathbb{N}} \bigl\Vert A_{i} \varLambda _{k}^{*}f_{k} \bigr\Vert ^{2}=\sum_{i\in\mathbb {N}} \Vert A_{i}P_{k}f \Vert ^{2}\leq b\sum _{i\in\mathbb{N}} \Vert A_{i}f \Vert ^{2}=0,$$

Now we give some equivalent characterizations for a g-frame to be a g-Riesz basis.

### Theorem 3.4

Let $$\{A_{i}\}$$ be a g-frame for H. Then the following are equivalent:

1. (1)

$$\{A_{i}\}$$ is a g-basis for H.

2. (2)

$$\{A_{i}\}$$ is g-w-linearly independent.

3. (3)

$$\{A_{i}\}$$ is a g-Riesz basis for H.

4. (4)

The g-R-dual $$\{{\mathcal{A}}_{i}\}$$ defined in Definition 2.2 is a g-Riesz basis for H.

5. (5)

If $$\lim_{n\rightarrow\infty}\sum_{i\in\mathbb{N}} \Vert {\mathcal{A}}_{i}x_{n} \Vert ^{2}=0$$, then $$\{ g_{i}\}=0$$, where $$x_{n}=\sum_{i=1}^{n}\varGamma_{i}^{*}g_{i}$$ for any $$n\in\mathbb{N}$$, $$\{g_{i}\}\in\bigoplus_{i\in \mathbb{N}}{H_{i}}$$.

6. (6)

$$\{A_{i}\}$$ is exact (i.e., if it ceases to be a g-frame whenever any one of its elements is removed), and the canonical dual g-frame is biorthonormal with $$\{ A_{i}\}$$.

### Proof

The equivalence of (1), (2), (3) can be obtained by [4, Lemma 2.16]. By [9, Corollary 2.6], we get the equivalence of (3) and (6). Since $$\{A_{i}\}$$ is a g-frame, we get $$\sum_{i\in\mathbb{N}} \Vert {\mathcal{A}}_{i}x_{n} \Vert ^{2}= \Vert \theta _{A}^{*}\theta_{\varGamma}x_{n} \Vert ^{2}$$. Then (5) holds if and only if $$\theta_{A}^{*}$$ is injective, i.e., (3) holds.

Similarly, by Definition 2.2, we have $$\theta_{\mathcal {A}}=\theta_{\varLambda}\theta_{A}^{*}\theta_{\varGamma}$$. For any $$f\in H$$, we obtain $$f\in\operatorname{ker}\theta_{\mathcal{A}}$$ if and only if $$\theta_{\varGamma}f\in\operatorname{ker}\theta_{A}^{*}$$. Thus we get the equivalence of (3), (4) by Theorem 2.6. □

## G-R-dual and the g-orthogonal sequence

### The characterization of g-R-dual

Let $$\{\varLambda_{i}\}$$ be a g-orthonormal basis for H. In this section we mainly investigate the conditions under which a g-Riesz sequence $$\{{\mathcal{A}}_{i}\}$$ is the g-R-dual of a g-frame $$\{A_{i}\}$$. We denote $$\{\widetilde{{\mathcal{A}}}_{i}\}$$ as the canonical dual g-frame of $$\{{\mathcal{A}}_{i}\}$$, which is also a g-Riesz sequence. Define $$C_{i}=A_{i}\theta_{\varLambda}^{*}\theta_{\widetilde{\mathcal {A}}}$$ for any $$i\in\mathbb{N}$$. Then

$$C^{*}_{i}g_{i}=\sum_{j\in\mathbb {N}} \widetilde{{\mathcal{A}}}_{j}^{*}\varLambda_{j}A_{i}^{*}g_{i}, \quad \forall g_{i}\in H_{i}.$$

Evidently, $$\{C_{i}\}$$ is a g-Bessel sequence for H. Let $$M=\operatorname {ran}\theta_{\mathcal{A}}^{*}$$. Thus $$\operatorname{ran}\theta _{C}^{*}\subset M$$. By Lemma 1.2, we also get $${\mathcal{A}}_{j}C^{*}_{i}=\varLambda _{j}A_{i}^{*}$$ for any $$i\in\mathbb{N}$$.

### Proposition 4.1

Let $$\{\varLambda_{i}\}$$ be a g-orthonormal basis for H, $$\{{\mathcal {A}}_{i}\}$$ be a g-Riesz basis for M, $$\{\widetilde{{\mathcal {A}}}_{i}\}$$ be the canonical dual g-frame of $$\{{\mathcal{A}}_{i}\}$$ in M, where M is a closed subspace of H. For any sequence $$\{ A_{i}\}$$, we have the following:

1. (1)

There exists a sequence $$\{\varGamma'_{i}\}$$ such that $$A_{i}=\varGamma'_{i}\theta_{\mathcal{A}}^{*}\theta_{\varLambda}$$ for any $$i\in\mathbb{N}$$, i.e., $$A^{*}_{i}g_{i}=\sum_{j\in \mathbb{N}}\varLambda_{j}^{*}{\mathcal{A}}_{j}{\varGamma'}_{i}^{*}g_{i}$$ for any $$g_{i}\in H_{i}$$.

2. (2)

The sequence $$\{\varGamma'_{i}\}$$ satisfying $$A_{i}=\varGamma'_{i}\theta_{\mathcal{A}}^{*}\theta_{\varLambda}$$ can be written as $$\varGamma'_{i}=C_{i}+D_{i}$$ for every $$i\in\mathbb{N}$$, where $$C_{i}=A_{i}\theta_{\varLambda}^{*}\theta_{\widetilde{\mathcal {A}}}$$, $$D_{i}\in B(H,H_{i})$$ and $$\operatorname{ran}D^{*}_{i}\subset M^{\bot}$$.

3. (3)

If $$H=M$$, the sequence $$\{\varGamma'_{i}\}$$ satisfying $$A_{i}=\varGamma'_{i}\theta_{\mathcal{A}}^{*}\theta_{\varLambda }$$ has the unique solution $$\varGamma'_{i}=C_{i}$$ for any $$i\in\mathbb {N}$$, where $$C_{i}=A_{i}\theta_{\varLambda}^{*}\theta_{\widetilde {\mathcal{A}}}$$.

### Proof

(1) Since $$A_{i}^{*}g_{i}=\sum_{j\in\mathbb{N}}\varLambda _{j}^{*}\varLambda_{j}A_{i}^{*}g_{i}$$ for any $$i\in\mathbb{N}$$, $$g_{i}\in H_{i}$$ and $${\mathcal{A}}_{j}C^{*}_{i}=\varLambda _{j}A_{i}^{*}$$, we have $$A_{i}^{*}g_{i}=\sum_{j\in\mathbb{N}}\varLambda _{j}^{*}{\mathcal{A}}_{j}C^{*}_{i}g_{i}$$. We take $$\varGamma'_{i}=C_{i}$$.

(2) For any $$i\in\mathbb{N}$$, take arbitrary operator $$D_{i}\in B(M^{\bot}, H_{i})$$. Obviously, $$\operatorname{ran}D^{*}_{i}\subset M^{\bot }$$ is satisfied. Let $$\varGamma'_{i}=C_{i}+D_{i}$$. Since $$M=\operatorname {ran}\theta_{\mathcal{A}}^{*}$$, by (1), we have

$$\varGamma'_{i}\theta_{\mathcal{A}}^{*} \theta_{\varLambda }=(C_{i}+D_{i})\theta_{\mathcal{A}}^{*} \theta_{\varLambda}=C_{i}\theta _{\mathcal{A}}^{*} \theta_{\varLambda}=A_{i}.$$

For the converse, suppose $$A_{i}=\varGamma'_{i}\theta_{\mathcal {A}}^{*}\theta_{\varLambda}$$ for any $$i\in\mathbb{N}$$. By (1), $$C_{i}\theta_{\mathcal{A}}^{*}\theta_{\varLambda}=A_{i}$$. Let $$D_{i}=\varGamma'_{i}-C_{i}$$. Hence, $$D_{i}\theta_{\mathcal {A}}^{*}\theta_{\varLambda}=0$$. Since $$M=\operatorname{ran}\theta_{\mathcal{A}}^{*}$$, $$M\subset\operatorname {ker}D_{i}$$. Thus $$\operatorname{ran}D^{*}_{i}\subset M^{\bot}$$.

(3) If $$H=M$$, we have $$D_{i}=0$$ for any $$i\in\mathbb{N}$$ from (2). □

Proposition 4.1 did not have any assumption on $$\{A_{i}\}$$ or use any relationship between $$\{A_{i}\}$$ and $$\{{\mathcal{A}}_{i}\}$$.

The next result exhibits that $$\{C_{i}\}$$ and $$\{A_{i}\}$$ have the common properties.

### Proposition 4.2

Let $$\{\varLambda_{i}\}$$ be a g-orthonormal basis for H, $$\{{\mathcal {A}}_{i}\}$$ be a g-Riesz basis for M with the frame bounds c and d, $$\{\widetilde{{\mathcal{A}}}_{i}\}$$ be the canonical dual g-frame of $$\{{\mathcal{A}}_{i}\}$$ in M, where M is a closed subspace of H. For a sequence $$\{A_{i}\}$$, define $$C_{i}=A_{i}\theta_{\varLambda }^{*}\theta_{\widetilde{\mathcal{A}}}$$, for any $$i\in\mathbb{N}$$, we have

1. (1)

If $$\{A_{i}\}$$ is a g-Bessel sequence for H with the upper bound b, then $$\{C_{i}\}$$ is a g-Bessel sequence for H with the upper bound $$bc^{-1}$$. Moreover, for any $$\{g_{i}\}\in\bigoplus_{i\in\mathbb {N}}{H_{i}}$$, we have

$$c \biggl\Vert \sum_{i\in\mathbb {N}}C^{*}_{i}g_{i} \biggr\Vert ^{2}\leq \biggl\Vert \sum_{i\in\mathbb {N}}A^{*}_{i}g_{i} \biggr\Vert ^{2}\leq d \biggl\Vert \sum _{i\in\mathbb {N}}C^{*}_{i}g_{i} \biggr\Vert ^{2}.$$

Specially, $$\{A_{i}\}$$ is g-w-linearly independent if and only if $$\{ C_{i}\}$$ is g-w-linearly independent.

2. (2)

If $$\{A_{i}\}$$ is a g-frame for H with the frame bounds $$a, b$$, then $$\{C_{i}\}$$ is a g-frame for M with the frame bounds $$ad^{-1}, bc^{-1}$$.

3. (3)

If $$\{A_{i}\}$$ is a g-Riesz basis for H with the frame bounds $$a, b$$, then $$\{C_{i}\}$$ is a g-Riesz basis for M with the frame bounds $$ad^{-1}, bc^{-1}$$.

4. (4)

If $$\{C_{i}\}$$ is a g-Bessel sequence for H with the upper bound $$b_{1}$$, then $$\{A_{i}\}$$ is a g-Bessel sequence for H with the upper bound $$b_{1}d$$.

5. (5)

If $$\{C_{i}\}$$ is a g-frame for M with the frame bounds $$a_{1}, b_{1}$$, then $$\{A_{i}\}$$ is a g-frame for H with the frame bounds $$a_{1}c, b_{1}d$$.

6. (6)

If $$\{C_{i}\}$$ is a g-Riesz basis for M with the frame bounds $$a_{1}, b_{1}$$, then $$\{A_{i}\}$$ is a g-Riesz basis for H with the frame bounds $$a_{1}c, a_{1}d$$.

### Proof

(1) Since $$C_{i}=A_{i}\theta_{\varLambda}^{*}\theta_{\widetilde{\mathcal {A}}}$$ for any $$i\in\mathbb{N}$$, for every $$f\in H$$, we have

$$\sum_{i\in\mathbb{N}} \Vert C_{i}f \Vert ^{2}=\sum_{i\in\mathbb{N}} \bigl\Vert A_{i}\theta_{\varLambda}^{*}\theta _{\widetilde{\mathcal{A}}}f \bigr\Vert ^{2}\leq bc^{-1} \Vert f \Vert ^{2}.$$

Moreover, because $$\theta_{C}^{*}=\theta_{\widetilde{\mathcal {A}}}^{*}\theta_{\varLambda}\theta_{A}^{*}$$, for any $$\{g_{i}\}\in \bigoplus_{i\in\mathbb{N}}{H_{i}}$$, we have

$$\biggl\Vert \sum_{i\in\mathbb {N}}C_{i}^{*}g_{i} \biggr\Vert ^{2}= \biggl\Vert \sum_{i\in\mathbb{N}} \widetilde {{\mathcal{A}}}_{i}^{*}\theta_{\varLambda} \theta_{A}^{*}g_{i} \biggr\Vert ^{2} \leq c^{-1} \biggl\Vert \sum_{i\in\mathbb{N}}A_{i}^{*}g_{i} \biggr\Vert ^{2}.$$

As $$\theta_{A}^{*}=\theta_{\varLambda}^{*}\theta_{\mathcal{A}}\theta _{C}^{*}$$, for every $$\{g_{i}\}\in\bigoplus_{i\in\mathbb {N}}{H_{i}}$$, we get

$$\biggl\Vert \sum_{i\in\mathbb {N}}A_{i}^{*}g_{i} \biggr\Vert ^{2}=\sum_{i\in\mathbb{N}} \bigl\Vert {\mathcal {A}}_{i}\theta_{C}^{*}g_{i} \bigr\Vert ^{2}\leq d \biggl\Vert \sum_{i\in\mathbb {N}}C_{i}^{*}g_{i} \biggr\Vert ^{2}.$$

Obviously, $$\{A_{i}\}$$ is g-w-linearly independent if and only if $$\{ C_{i}\}$$ is g-w-linearly independent from the above.

(2) The case of upper bound was obtained by (1). Similarly as (1), for every $$f\in M$$, we get

$$ad^{-1} \Vert f \Vert ^{2}\leq a \bigl\Vert \theta_{\varLambda}^{*}\theta _{\widetilde{\mathcal{A}}}f \bigr\Vert ^{2}\leq\sum_{i\in\mathbb {N}} \bigl\Vert A_{i}\theta_{\varLambda}^{*}\theta_{\widetilde{\mathcal {A}}}f \bigr\Vert ^{2}=\sum_{i\in\mathbb{N}} \Vert C_{i}f \Vert ^{2}.$$

(3) Suppose $$\{A_{i}\}$$ is a g-Riesz basis for H. Since $$\{C_{i}\}$$ is a g-frame for M by (2) and is g-w-linearly independent by (1), $$\{C_{i}\}$$ is a g-Riesz basis for M by [4, Lemma 2.16]. The frame bounds can be obtained by (2).

The rest is similar to the above. □

From the above, $$\{C_{i}\}$$, $$\{A_{i}\}$$ have the same properties, but the bounds may not be common.

### Corollary 4.3

Let $$\{\varLambda_{i}\}$$ be a g-orthonormal basis for H, $$\{{\mathcal{A}}_{i}\}$$ be a g-orthonormal basis for M, where M is a closed subspace of H. For a sequence $$\{A_{i}\}$$, define $$C_{i}=A_{i}\theta_{\varLambda}^{*}\theta_{\widetilde{\mathcal{A}}}$$ for any $$i\in\mathbb{N}$$, we have:

1. (1)

$$\{C_{i}\}$$ is a g-Bessel sequence for H if and only if $$\{A_{i}\}$$ is a g-Bessel sequence for H with the same bound.

2. (2)

$$\{C_{i}\}$$ is a g-frame for M if and only if $$\{ A_{i}\}$$ is a g-frame for H with the same bounds.

3. (3)

$$\{C_{i}\}$$ is a g-Riesz basis for M if and only if $$\{A_{i}\}$$ is a g-Riesz basis for H with the same bounds.

### Proof

Take $$c=d=1$$ by the proof of Proposition 4.2, which can be obtained directly. □

Let $$\{{\mathcal{A}}_{i}\}$$ be a g-Riesz basis for M, where M is a closed subspace of H. Let $${\mathscr{A}}_{i}={\mathcal {A}}_{i}S_{\mathcal{A}}^{-\frac{1}{2}}$$ for any $$i\in\mathbb{N}$$, where $$S_{\mathcal{A}}$$ is the frame operator of $$\{{\mathcal{A}}_{i}\}$$. Then $$\{{\mathscr{A}}_{i}\}$$ is a g-orthonormal basis for M. Let $$\{\varLambda_{i}\}$$ be a g-orthonormal basis for H and $$\varTheta=\theta _{\varLambda}^{*}\theta_{\mathscr{A}}$$. Obviously, $$\varTheta: M\rightarrow H$$ is unitary and $${\mathscr{A}}_{i}=\varLambda_{i}\varTheta$$. Then we have the following result.

### Proposition 4.4

Let $$\{\varLambda_{i}\}$$ be a g-orthonormal basis for H, $$\{{\mathcal {A}}_{i}\}$$ be a g-Riesz basis for M with the frame bounds $$c, d$$, where M is a closed subspace of H, $$\{A_{i}\}$$ be a g-frame for H with the frame bounds $$a, b$$. Define $$C_{i}=A_{i}\theta_{\varLambda }^{*}\theta_{\widetilde{\mathcal{A}}}$$ for every $$i\in\mathbb{N}$$. Then the following are equivalent:

1. (1)

$$\{C_{i}\}$$ is a Parseval g-frame for M.

2. (2)

$$S_{\mathcal{A}}=\varTheta^{*}S_{A}\varTheta$$, where $$\varTheta=\theta_{\varLambda}^{*}\theta_{\widetilde{\mathcal {A}}}S_{\mathcal{A}}^{\frac{1}{2}}$$.

### Proof

By Proposition 4.2, $$\{C_{i}\}$$ is a g-frame for M. Since $$\theta_{C}=\theta_{A}\theta_{\varLambda}^{*}\theta_{\widetilde {\mathcal{A}}}$$ and $$\theta_{\widetilde{\mathcal{A}}}=\theta_{\varLambda }\varTheta S_{\mathcal{A}}^{-\frac{1}{2}}$$, we have $$S_{C}=S_{\mathcal{A}}^{-\frac{1}{2}}\varTheta^{*}S_{A}\varTheta S_{\mathcal {A}}^{-\frac{1}{2}}$$. Obviously, $$S_{C}=P$$ if and only if $$S_{\mathcal{A}}=\varTheta ^{*}S_{A}\varTheta$$, where P is the orthogonal projection from H onto M. □

If $$\{A_{i}\}$$ is a tight g-frame for H with the bound a. Let $$\{ {\mathcal{A}}_{i}\}$$ be a tight g-Riesz basis for M with frame bound a. Then $$S_{A}=aI$$, $$S_{\mathcal{A}}=aP$$. Thus Proposition 4.4(2) holds obviously. Then we get Corollary 4.6 directly.

### Proposition 4.5

Let $$\{\varLambda_{i}\}$$ be a g-orthonormal basis for H, $$\{{\mathcal {A}}_{i}\}$$ be a g-Riesz basis for M, where M is a closed subspace of H. If $$\{A_{i}\}$$ is a g-frame for H, define $$C_{i}=A_{i}\theta _{\varLambda}^{*}\theta_{\widetilde{\mathcal{A}}}$$ for any $$i\in\mathbb {N}$$. Then the following are equivalent:

1. (1)

If $$\{{\mathcal{A}}_{i}\}$$ is the g-R-dual sequence of $$\{A_{i}\}$$ with respect to two g-orthonormal bases $$\{\varLambda_{i}\}$$, $$\{\varGamma_{i}\}$$.

2. (2)

There exists a g-orthonormal basis $$\{\varGamma_{i}\}$$ for H such that $$A_{i}=\varGamma_{i}\theta_{\mathcal{A}}^{*}\theta _{\varLambda}$$ for every $$i\in\mathbb{N}$$.

3. (3)

There exists a g-orthonormal basis $$\{\varGamma_{i}\}$$ for H such that $$C_{i}=\varGamma_{i}P$$ for every $$i\in\mathbb{N}$$, where P is the orthogonal projection from H onto M.

4. (4)

$$\{C_{i}\}$$ is a Parseval g-frame for M and $$\operatorname{dim}\operatorname{ker}\theta_{C}^{*}=\operatorname{dim}M^{\bot}$$.

5. (5)

$$S_{\mathcal{A}}=\varTheta^{*}S_{A}\varTheta$$ and $$\operatorname{dim}\operatorname{ker}\theta_{C}^{*}=\operatorname{dim}M^{\bot}$$, where $$\varTheta=\theta_{\varLambda}^{*}\theta_{\widetilde{\mathcal {A}}}S_{\mathcal{A}}^{\frac{1}{2}}$$.

### Proof

(1) (2) By Definition 2.2, we have $${\mathcal {A}}_{i}^{*}=\theta_{\varGamma}^{*}\theta_{A}\varLambda_{i}^{*}$$ for every $$i\in\mathbb{N}$$. Hence, $$A_{i}=\varGamma_{i}\theta_{\mathcal {A}}^{*}\theta_{\varLambda}$$.

(2) (1) It is obvious by Definition 2.2. The equivalence of (2) and (3) can be obtained by Proposition 4.1.

(3) (4) For any $$\{g_{i}\}\in\bigoplus_{i\in \mathbb{N}}{H_{i}}$$, we have

$$\theta_{C}^{*}\{g_{i}\}=\sum _{i\in\mathbb {N}}C_{i}^{*}g_{i}=\sum _{i\in\mathbb{N}}P\varGamma _{i}^{*}g_{i}=P \theta_{\varGamma}^{*}\{g_{i}\}.$$

Obviously, $$\{g_{i}\}\in\operatorname{ker}\theta_{C}^{*}$$ if and only if $$\theta_{\varGamma}^{*}\{g_{i}\}\in M^{\bot}$$. Then $$\operatorname {dim}\operatorname{ker}\theta_{C}^{*}=\operatorname{dim}M^{\bot}$$ as $$\theta _{\varGamma}$$ is unitary. Evidently, $$\{C_{i}\}$$ is a Parseval g-frame for M.

(4) (3) Suppose $$\{C_{i}\}$$ is a Parseval g-frame for M. Let $$K=M\oplus(\operatorname{ran}\theta_{C})^{\bot}$$, $$T_{i}=C_{i}\oplus P_{i}Q^{\bot}$$ for any $$i\in\mathbb{N}$$, where $$Q, P_{i}$$ are the orthogonal projection from $$\bigoplus_{i\in \mathbb{N}}{H_{i}}$$ onto $$\operatorname{ran}\theta_{C}$$, $$H_{i}$$, respectively, for every $$i\in\mathbb{N}$$. It is easy to get $$\{T_{i}\}$$ is a g-orthonormal basis for K by [7, Theorem 4.1].

Since $$\operatorname{dim}\operatorname{ker}\theta_{C}^{*}=\operatorname{dim}M^{\bot }$$, there exists a unitary operator $$V: M^{\bot}\rightarrow\operatorname {ker}\theta_{C}^{*}$$. Let $$\varGamma_{i}=T_{i}(P\oplus V)=C_{i}\oplus P_{i}Q^{\bot}V$$ for every $$i\in\mathbb{N}$$. As $$P\oplus V: M\oplus M^{\bot}\rightarrow M\oplus(\operatorname{ran}\theta _{C})^{\bot}$$ is unitary, where P is the orthogonal projection from H onto M, we see that $$\{\varGamma_{i}\}$$ is a g-orthonormal basis for H by [6, Theorem 3.5]. Obviously, we have $$C_{i}=\varGamma_{i}P$$. The equivalence of (4), (5) is direct by Proposition 4.4. □

By Proposition 4.5, we can also get the following corollary, which was showed in [3, Theorem 2.7].

### Corollary 4.6

Let $$\{\varLambda_{i}\}$$ be a g-orthonormal basis for H, $$\{{\mathcal{A}}_{i}\}$$ be a tight g-Riesz basis for M with the frame bound a, where M is a closed subspace of H. If $$\{A_{i}\}$$ is a tight g-frame with the frame bound a. Then there exists a g-orthonormal basis $$\{\varGamma_{i}\}$$ for H such that $$\{{\mathcal{A}}_{i}\}$$ is the g-R-dual of $$\{A_{i}\}$$ with respect to two g-orthonormal bases $$\{\varLambda_{i}\}$$, $$\{\varGamma_{i}\}$$ if and only if $$\operatorname{dim}\operatorname{ker}\theta_{C}^{*}=\operatorname {dim}M^{\bot}$$, where $$C_{i}=A_{i}\theta_{\varLambda}^{*}\theta _{\widetilde{\mathcal{A}}}$$ for any $$i\in\mathbb{N}$$.

### Proof

By Proposition 4.2(3), $$\{C_{i}\}$$ is a Parseval g-frame for M. It is obvious by Proposition 4.5. □

### Corollary 4.7

Let $$\{\varLambda_{i}\}$$ be a g-orthonormal basis for H, $$\{{\mathcal{A}}_{i}\}$$ be a g-Riesz basis for M, $$\{\widetilde {{\mathcal{A}}}_{i}\}$$ be the canonical dual g-frame of $$\{{\mathcal {A}}_{i}\}$$ in M, where M is a closed subspace of H. If $$\{A_{i}\}$$ is a g-frame for H. Define $$C_{i}=A_{i}\theta_{\varLambda}^{*}\theta _{\widetilde{\mathcal{A}}}$$ for any $$i\in\mathbb{N}$$. For any $$\{ g_{i}\}\in\bigoplus_{i\in\mathbb{N}}{H_{i}}$$, let $$g=\theta_{\varLambda}^{*}\{g_{i}\}\in H$$, $$h=\theta_{\mathcal{A}}^{*}\{ g_{i}\}\in M$$. Then there exists a g-orthonormal basis $$\{\varGamma_{i}\}$$ for H such that $$\{{\mathcal{A}}_{i}\}$$ is the g-R-dual of $$\{ A_{i}\}$$ with respect to two g-orthonormal bases $$\{\varLambda_{i}\}$$, $$\{ \varGamma_{i}\}$$ if and only if $$\sum_{i\in\mathbb {N}} \Vert A_{i}g \Vert ^{2}= \Vert h \Vert ^{2}$$ and $$\operatorname{dim}\operatorname{ker}\theta _{C}^{*}=\operatorname{dim}M^{\bot}$$.

### Proof

Obviously, we have

$$\sum_{i\in\mathbb{N}} \Vert A_{i}g \Vert ^{2}= \bigl\Vert \theta _{A}\theta_{\varLambda}^{*} \{g_{i}\} \bigr\Vert ^{2}= \bigl\Vert \theta_{\mathcal{A}}^{*}\{ g_{i}\} \bigr\Vert ^{2}= \Vert h \Vert ^{2}.$$

The result now follows from Proposition 4.5 directly. □

### The construction of orthogonal sequence

Now we will construct a sequence $$\{\varGamma'_{i}\}$$ such $$A_{i}=\sum_{j\in\mathbb{N}}\varGamma'_{i}\widetilde{{\mathcal{A}}}_{j}^{*}\varLambda _{j}$$, which is characterized in Proposition 4.1.

### Proposition 4.8

Let $$\{\varLambda_{i}\}$$ be a g-orthonormal basis for H, $$\{{\mathcal {A}}_{i}\}$$ be a g-Riesz basis for M, $$\{\widetilde{{\mathcal {A}}}_{i}\}$$ be the canonical dual g-frame of $$\{{\mathcal{A}}_{i}\}$$ in M, where M is a closed subspace of H. If $$\operatorname{dim}M^{\bot }=\sum_{i}\operatorname{dim}H_{i}=\infty$$, we have:

1. (1)

For any sequence $$\{A_{i}\}$$, there exists a g-w-linearly independent sequence $$\{\varGamma'_{i}\}$$ such that $$A_{i}=\sum_{j\in\mathbb{N}}\varGamma'_{i}\widetilde{{\mathcal {A}}}_{j}^{*}\varLambda_{j}$$ for every $$i\in\mathbb{N}$$.

2. (2)

For any g-Bessel sequence $$\{A_{i}\}$$, there exists a norm-bounded and g-w-linearly independent sequence $$\{\varGamma'_{i}\}$$ such that $$A_{i}=\sum_{j\in\mathbb{N}}\varGamma'_{i}\widetilde {{\mathcal{A}}}_{j}^{*}\varLambda_{j}$$ for every $$i\in\mathbb{N}$$.

3. (3)

For any operator sequence $$\{A_{i}\}$$, there exists a g-orthogonal sequence $$\{\varGamma'_{i}\}$$ such that $$A_{i}=\sum_{j\in\mathbb{N}}\varGamma'_{i}\widetilde{{\mathcal {A}}}_{j}^{*}\varLambda_{j}$$ for every $$i\in\mathbb{N}$$.

### Proof

(1) Since $$\operatorname{dim}M^{\bot}=\sum_{i\in\mathbb{N}}\operatorname {dim}H_{i}$$, there exists a g-orthonormal basis $$\{E_{i}\}$$ for $$M^{\bot}$$ by [5, Theorem 3.1] with $$E_{i}\in B(M^{\bot}, H_{i})$$ for any $$i\in\mathbb{N}$$. Let $$W_{i}=\overline{\operatorname {ran}} {E_{i}^{*}}$$ for any $$i\in\mathbb{N}$$. Then $$M^{\bot}= \bigoplus_{i\in\mathbb{N}}W_{i}$$ and $$E_{i}: W_{i}\rightarrow H_{i}$$ is unitary. Let $$C_{i}=A_{i}\theta_{\varLambda}^{*}\theta _{\widetilde{\mathcal{A}}}$$ for any $$i\in\mathbb{N}$$. Then $${\mathcal {A}}_{i}E_{j}^{*}=0$$ and $$C_{i}E_{j}^{*}=\sum_{k\in\mathbb {N}}A_{i}\varLambda_{k}^{*}\widetilde{{\mathcal{A}}}_{k}E_{j}^{*}=0$$.

Since there exists an invertible operator $$D_{i}: W_{i}\rightarrow H_{i}$$ for any $$i\in\mathbb{N}$$, we see that $$D_{i}E_{i}^{*}+C_{i}E_{i}^{*}=D_{i}E_{i}^{*}\in B(H,H_{i})$$ is invertible. Let $$\varGamma'_{i}=D_{i}+C_{i}\in B(H, H_{i})$$. Obviously, $$\varGamma'_{i}\neq0$$.

For any $$\{g_{i}\}\in\bigoplus_{i\in\mathbb{N}}H_{i}$$, if $$\sum_{i\in\mathbb{N}}{\varGamma}_{i}^{'*}g_{i}=0$$, then, for any $$j\in\mathbb{N}$$, we have

$$E_{j}\sum_{i\in\mathbb{N}}{\varGamma '}_{i}^{*}g_{i}=\sum _{i\in\mathbb {N}} \bigl(E_{j}C_{i}^{*}+E_{j}D_{i}^{*} \bigr)g_{i}=E_{j}D_{j}^{*}g_{j}=0.$$

Then $$g_{j}=0$$.

(2) By the proof of (1), we can choose $$D_{i}$$ such that $$\Vert D_{i} \Vert =1$$ (if not, we choose $$D'_{i}= \frac{D_{i}}{ \Vert D_{i} \Vert }$$) for any $$i\in\mathbb{N}$$. By Proposition 4.2, $$\{C_{i}\}$$ is a g-Bessel sequence for M. Suppose the upper bound of $$\{C_{i}\}$$ is b. Then $$\Vert C_{i} \Vert \leq b$$. Hence, for every $$i\in\mathbb{N}$$, $$g_{i}\in H_{i}$$, we have

$$\bigl\Vert {\varGamma '}_{i}^{*}g_{i} \bigr\Vert ^{2}= \bigl\Vert C^{*}_{i}g_{i} \bigr\Vert ^{2}+ \bigl\Vert D^{*}_{i}g_{i} \bigr\Vert ^{2}\leq \bigl(b^{2}+1 \bigr) \Vert g_{i} \Vert ^{2}.$$

(3) By Proposition 4.1, the sequence $$\{\varGamma'_{i}\}$$ such that $$A_{i}=\sum_{j\in\mathbb{N}}\varGamma'_{i}\widetilde {{\mathcal{A}}}_{j}^{*}\varLambda_{j}=\varGamma'_{i}\theta_{\widetilde {{\mathcal{A}}}}^{*}\theta_{\varLambda}$$ can be written as $$\varGamma '_{i}=C_{i}+D_{i}$$, where $$C_{i}=A_{i}\theta_{\varLambda}^{*}\theta _{\widetilde{\mathcal{A}}}$$, $$\overline{\operatorname{ran}} D^{*}_{i}\subset M^{\bot}$$ for any $$i\in\mathbb{N}$$. For every $$i, j\in\mathbb{N}, i\neq j$$, $$g_{i}\in H_{i}$$, $$g_{j}\in H_{j}$$, we have

$$\bigl\langle {\varGamma'}_{i}^{*}g_{i}, {\varGamma '_{j}}^{*}g_{j} \bigr\rangle =0 \quad\text{if and only if} \quad \bigl\langle C_{i}^{*}g_{i}, C^{*}_{j}g_{j} \bigr\rangle + \bigl\langle D_{i}^{*}g_{i}, D^{*}_{j}g_{j} \bigr\rangle =0.$$

We will use the following inductive procedure to construct $$\{D_{i}\}$$ such that $$\overline{\operatorname{ran}} D^{*}_{i}\subset M^{\bot}$$ and $$D_{j}D^{*}_{i}=-C_{j}C^{*}_{i}$$ for every $$i,j\in\mathbb{N}$$, $$i\neq j$$. Let $$T_{ij}=-C_{i}C^{*}_{j}\in B(H_{j},H_{i})$$. Then $$T_{ij}^{*}=T_{ji}$$. Let $$I_{i}$$ be the identity on $$H_{i}$$.

(1) Let $$D_{1}^{*}=E_{1}^{*}$$.

(2) Let $$D_{2}^{*}=E^{*}_{1}X_{1,2}^{*}+E_{2}^{*}$$, where $$X_{1,2}^{*}=T_{12}$$.

Obviously, $$D_{1}D_{2}^{*}=E_{1}E^{*}_{1}X_{1,2}^{*}+E_{1}E_{2}^{*}=T_{12}$$. Then $$\varGamma'_{1}{\varGamma'_{2}}^{*}=0$$.

3) For any $$k\in\mathbb{N}$$, assuming that we have gotten operators $$D_{1}, D_{2}, \ldots, D_{k}$$ in terms of $$X_{i,k}\in B(H_{i},H_{k})$$ ($$i=1,\ldots, k-1$$) such that $$D_{k}^{*}=\sum_{i=1}^{k-1}E^{*}_{i}X_{i,k}^{*}+E_{k}^{*}$$. Then, for $$k+1$$, we define $$D_{k+1}$$ by $$D_{k+1}^{*}=\sum_{i=1}^{k}E^{*}_{i}X_{i,k+1}^{*}+E_{k+1}^{*}$$, where operators $$X_{i,k+1}\ (i=1,2,\ldots,k)$$ are given by the following equation:

$\left(\begin{array}{cccc}{I}_{1}& & & \\ {X}_{12}& {I}_{2}& & \\ ⋮& & \ddots & \\ {X}_{1k}& {X}_{2k}& \cdots & {I}_{k}\end{array}\right)\left(\begin{array}{c}{X}_{1,k+1}^{\ast }\\ {X}_{2,k+1}^{\ast }\\ ⋮\\ {X}_{k,k+1}^{\ast }\end{array}\right)=\left(\begin{array}{c}{T}_{1,k+1}\\ {T}_{2,k+1}\\ ⋮\\ {T}_{k,k+1}\end{array}\right).$

Obviously, we can obtain $$X_{i,k+1}\in B(H_{i},H_{k+1})$$ ($$i=1,\ldots, k$$). Thus we have constructed the sequence $$\{D_{i}\}$$ and obtained $$\{ \varGamma'_{i}\}$$ by $$\varGamma'_{i}=C_{i}+D_{i}$$ for any $$i\in\mathbb {N}$$. Then $$\{\varGamma'_{i}\}$$ such that $$\varGamma'_{i}{\varGamma '}_{j}^{*}=0$$ for every $$i,j\in\mathbb{N}$$ with $$i\neq j$$.

Lastly, we show the sequence $$\{\varGamma_{i}'\}$$ satisfies the desired condition: $$A_{i}=\sum_{j\in\mathbb{N}}\varGamma_{i}'\mathcal {A}_{j}^{*}\varLambda_{j}$$ for all $$i\in\mathbb{N}$$.

Since $$(\operatorname{ker}D_{i})^{\bot}=\overline{\operatorname{ran}} D^{*}_{i}\subset M^{\bot}$$ and $$\overline{\operatorname{ran}} \widetilde{{\mathcal {A}}}_{j}^{*}\subset M$$ for any $$i, j\in\mathbb{N}$$, we have

$$\overline{\operatorname{ran}} \widetilde{{\mathcal {A}}}_{j}^{*} \subset M\subset\operatorname{ker}D_{i}.$$

Hence, $$D_{i}\widetilde{{\mathcal{A}}}_{j}^{*}=0$$ for any $$i, j\in \mathbb{N}$$. On the other hand, since $$C_{i}=A_{i}\theta_{\varLambda }^{*}\theta_{\widetilde{\mathcal{A}}}$$ for any $$i\in\mathbb{J}$$, we get $${\mathcal{A}}_{j}C^{*}_{i}=\varLambda_{j}A_{i}^{*}$$. By $$A_{i}^{*}g_{i}=\sum_{j\in\mathbb{N}}\varLambda_{j}^{*}\varLambda _{j}A_{i}^{*}g_{i}$$ for any $$g_{i}\in H_{i}$$, any $$i\in\mathbb{N}$$, we have $$A_{i}^{*}g_{i}=\sum_{j\in\mathbb{N}}\varLambda _{j}^{*}{\mathcal{A}}_{j}C^{*}_{i}g_{i}$$. So $$\sum_{j\in \mathbb{N}}C_{i}\widetilde{{\mathcal{A}}}_{j}^{*}\varLambda_{j}=A_{i}$$ for any $$i\in\mathbb{N}$$. Then

$$\sum_{j\in\mathbb{N}}\varGamma'_{i} \widetilde {{\mathcal{A}}}_{j}^{*}\varLambda_{j}= \sum_{j\in\mathbb {N}}(C_{i}+D_{i}) \widetilde{{\mathcal{A}}}_{j}^{*}\varLambda_{j}= \sum_{j\in\mathbb{N}}C_{i}\widetilde{{ \mathcal{A}}}_{j}^{*}\varLambda _{j}=A_{i}, \quad \forall i\in\mathbb{N}.$$

□

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### Acknowledgements

The authors would like to express their gratitude to the reviewers for their helpful comments and suggestions.

Not applicable.

## Funding

This work is supported by National Natural Science Foundation of China (Nos. 11671201 and 11771379).

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

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Correspondence to Liang Li.

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Li, L., Li, P. Characterizing the R-duality of g-frames. J Inequal Appl 2019, 69 (2019). https://doi.org/10.1186/s13660-019-2022-x

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• DOI: https://doi.org/10.1186/s13660-019-2022-x

• 46L10
• 42C40
• 42C15

### Keywords

• Frames
• G-frames
• G-R-duals
• G-orthonormal bases
• Dilations
• G-duals
• G-Riesz sequences