G-Frame and Riesz Sequences in Hilbert Spaces

G-frames generalize frames in Hilbert spaces. The literatures show that g-frames and frames share many similar properties, while they behave differently in redundancy and perturbation properties. Interestingly, g-frames have been extensively studied, but g-frame sequences have not. This problem is nontrivial since a g-frame and a frame both involve all vectors in the same Hilbert space, while a g-frame sequence and a frame sequence do not. They involve different linear spans. Using the synthesis and Gram matrix methods, we in this paper characterize g-frame sequences and g-Riesz sequences; obtain the Pythagorean theorem for g-orthonormal systems. These results recover several known results and lead to some new results on g-frames.     

Hilbert空间中g-Parseval框架的一些性质

在Hilbert空间中讨论g-Parseval框架的一些性质,得到g-Parseval框架的一些恒等式和不等式.     

Hilbert 空间中的广义正交基

广义正交基是Hilbert 空间中正交基的一个自然推广. 本文首先给出一个广义正交基存在的较弱的充要条件; 然后研究广义正交基的性质, 特别地, 得到广义正交基版本的一些有关正交基的经典性质, 如广义正交基的Bessel 等式和不等式等. 作为广义正交基的一个应用, 本文给出广义Riesz 基的一些新刻画. 最后本文讨论广义框架的冗余问题.     

Continuous g-frame and g-Riesz sequences in Hilbert spaces

A continuous g-frame is a generalization of g-frames and continuous frames, but they behave much differently from g-frames due to the underlying characteristic of measure spaces. Now, continuous g-frames have been extensively studied, while continuous g-sequences such as continuous g-frame sequence, g-Riesz sequences, and continuous g-orthonormal systems have not. This paper addresses continuous g-sequences. It is a continuation of Zhang and Li, in Numer. Func. Anal. Opt., 40 (2019), 1268-1290, where they dealt with g-sequences. In terms of synthesis and Gram operator methods, we in this paper characterize continuous g-Bessel, g-frame, and g-Riesz sequences, respectively, and obtain the Pythagorean theorem for continuous g-orthonormal systems. It is worth that our results are similar to the case of g-ones, but their proofs are nontrivial. It is because the definition of continuous g-sequences is different from that of g-sequences due to it involving general measure space.     

Joint Similarities and Parameterizations for Dilations of Dual g -frame Pairs in Hilbert Spaces

In this paper, we give an operator parameterization for the set of dilations of a given pair of dual g-frames and the set of dilations of pairs of dual g-frames of a given g-frame. In particular, for the dilations of a given pair of dual g-frames, we introduce the concept of joint complementary g-frames and prove that the joint complementary g-frames of a pair of dual g-frames are unique in the sense of joint similarity, which then helps to obtain a sufficient condition such that the complementary g-frames of a g-frame are unique in the sense of similarity and show that the set of dilations of a given dual g-frame pair are parameterized by a set of invertible diagonal operators. For the dilations of pairs of dual g-frames, we prove that the set of dilations of pairs of dual g-frames are parameterized by a set of invertible upper triangular operators.     

g-Banach框架的稳定性研究(英文)

Abdollahpour引入了g-Banach框架的概念,本文研究了g-Banach框架的扰动和稳定性,得到了关于g-Banach框架的一些稳定性条件,Abdollahpour文献中有关稳定性结果是本文结果的特例.     

Approximate duality of g-frames in Hilbert spaces

In this article, we introduce and characterize approximate duality for g-frames. We get some important properties and applications of approximate duals. We also obtain some new results in approximate duality of frames, and generalize some of the known results in approximate duality of frames to g-frames. We also get some results for fusion frames, and perturbation of approximately dual g-frames. We show that approximate duals are stable under small perturbations and they are useful for erasures and reconstruction.     

Hilbert空间中的g-Bessel序列的一些等式与不等式(II)

Hilbert 空间中的g- 框架是框架的自然推广, 它们包含了许多推广的框架, 如子空间框架或fusion 框架、斜框架和拟框架等. 它们有许多与框架类似的性质, 但是并不是所有的性质都是相似的.例如, 无冗框架等价于Riesz 基, 但是无冗g- 框架不等价于g-Riesz 基. 一些作者将Hilbert 空间中的框架和对偶框架的等式和不等式推广到g- 框架和对偶g- 框架. 本文建立Hilbert 空间中的g-Bessel序列或g- 框架的一些新的等式和不等式. 本文还给出这些不等式的等号成立的充要条件. 这些结果推广和改进了由Balan, Casazza 和G?vruta 等得到的著名结果.     

G-Frame Representation and Invertibility of G-Bessel Multipliers

In this paper we show that every g-frame for an infinite dimensional Hilbert space H can be written as a sum of three g-orthonormal bases for H. Also, we prove that every g-frame can be represented as a linear combination of two g-orthonormal bases if and only if it is a g-Riesz basis. Further, we show each g-Bessel multiplier is a Bessel multiplier and investigate the inversion of g-frame multipliers. Finally, we introduce the concept of controlled g-frames and weighted g-frames and show that the sequence induced by each controlled g-frame (resp., weighted g-frame) is a controlled frame (resp., weighted frame).     

A New Characterization on g-frames in Hilbert C·-Modules

In this note, we establish a new characterization on g-frames in Hilbert C~*-modules from the operator-theoretic point of view, with which we provide a correction to one result recently obtained by Yao(Yao X Y. Some properties of g-frames in Hilbert C~*-modules(in Chinese). Acta Math. Sinica, 2011, 54(1): 1–8.).