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. (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].

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. (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. □

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. (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

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

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. (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. □

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. (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. □

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. (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:

( I 1 X 12 I 2 ⋮ ⋱ X 1 k X 2 k ⋯ I k ) ( X 1 , k + 1 ∗ X 2 , k + 1 ∗ ⋮ X k , k + 1 ∗ ) = ( T 1 , k + 1 T 2 , k + 1 ⋮ T k , k + 1 ) .

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

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

 □