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.
Highlights
1 Introduction Duality principles in Gabor theory play a fundamental role in analyzing the Gabor system
We are interested in the duality principles for g-frames
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
Summary
Duality principles in Gabor theory play a fundamental role in analyzing the Gabor system. In the following results we show the properties of g-R-dual in the case that {Ai} is a g-frame sequence by the corresponding analysis operators. Lemma 2.7 Let {Ai}, {Bi} be two g-frames for H, {Ai}, {Bi} be their g-R-dual sequences defined in Definition 2.2, respectively. ΘA∗θB = I if and only if θAθB∗ = I i∈N Hi , i.e., A∗i gi, Bj∗gj = δij gi, gj for any i, j ∈ N, any gi ∈ Hi, gj ∈ Hj. The following shows that the g-R-dual of the canonical dual g-frame is the “minimal” and has the “smallest distance” with {Ai} 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 3.2 Let {Ai} be a g-Bessel sequence for H, {Ai} defined in Definition 2.2 be its g-R-dual.
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