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‐
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.
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.
In this paper, we only interested in the case when the g-orthonormal basis for H exists, which is equivalent to the following result.
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.
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.
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].
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 \(\{{\mathcal{A}}_{i}\}\).
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].
The following result was given in [
3, Theorem 4.1], we here give a simple illustration by the use of the analysis operators.
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].
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
\(\{A_{i}\}\).
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
\(\{A_{i}\}\).
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
\(\{{\mathcal{A}}_{i}\}\).
Now we give some equivalent characterizations for a g-frame to be a g-Riesz basis.
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