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.

1 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) $\{A_{i}\}$ 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) $\{A_{i}\}$ 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].

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

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

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

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

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

• (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) $\Vert {\mathcal{B}}_{i}^{*}g_{i}-{\mathcal {A}}_{i}^{*}g_{i} \Vert \leq \Vert {\mathcal{C}}_{i}^{*}g_{i}-{\mathcal {A}}_{i}^{*}g_{i} \Vert$ 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. □

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

• (2) $\{{\mathcal{A}}_{i}\}$ is g-w-linearly independent.

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

• (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) 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

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

which is a contradiction. □

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

• (2) $\{A_{i}\}$ is g-w-linearly independent.

• (3) $\{A_{i}\}$ is a g-Riesz basis for H.

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

• (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) $\{A_{i}\}$ 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. □

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

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

• (3) $\{C_{i}\}$ 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) $\{C_{i}\}$ is a Parseval g-frame for M.

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

• (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. □

4.2 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) 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) 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) 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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