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.     

Exact g-frames in Hilbert spaces

G-frames, which were considered recently as generalized frames in Hilbert spaces, have many properties similar to those of frames, but not all the properties are similar. For example, exact frames are equivalent to Riesz bases, but exact g-frames are not equivalent to g-Riesz bases. In this paper, we firstly give a characterization of an exact g-frame in a complex Hilbert space. We also obtain an equivalent relation between an exact g-frame and a g-Riesz basis under some conditions. Lastly we consider the stability of an exact g-frame for a Hilbert space under perturbation. These properties of exact g-frames for Hilbert spaces are not similar to those of exact frames.     

Stability of g-frames

g-Frames are natural generalizations of frames which cover many other recent generalizations of frames, e.g., bounded quasi-projectors, frames of subspaces, outer frames, oblique frames, pseudo-frames and a class of time-frequency localization operators. Moreover, it is known that g-frames are equivalent to stable space splittings. In this paper, we study the stability of g-frames. We first present some properties for g-Bessel sequences. Then we prove that g-frames are stable under small perturbations. We also study the stability of dual g-frames.     

广义框架的不相交性

Hilbert空间中框架的不相交性是由Han和Larson首先提出的,它与超框架有着密切的联系,在超框架及框架的构造中扮演重要的角色.广义框架是通常框架的推广.该文利用超广义框架刻画了广义框架的不相交性、强不相交性及弱不相交性;借助所得结果,给出了已知定理的不同证明方法;建立了对偶广义框架之间强不相交相和弱不相交性之间的关系;最后利用给定的强不相交的广义框架构造新的(超)对偶广义框架,所得结论恢复已有结果.     

SOME PROPERTIES OF OPERATOR-VALUED FRAMES

Operator-valued frames(or g-frames) are generalizations of frames and fusion frames and have been used in packets encoding, quantum computing, theory of coherent states and more. In this article, we give a new formula for operator-valued frames for finite dimensional Hilbert spaces. As an application, we derive in a simple manner a recent result of A. Najati concerning the approximation of g-frames by Parseval ones. We obtain also some results concerning the best approximation of operator-valued frames by its alternate duals,with optimal estimates.     

Hibert空间中广义框架的非交性

引入了Hilbert空间H中广义框架的非交性、强非交性,讨论了它们的一些性质;并且引入了保非交算子、强保非交算子,证明了酉算子、可逆算子是强保非交算子,下有界算子、余等距算子是保非交算子.     

加权Morrey-Herz空间上的广义Hausdorff算子

引入了一种新的广义Hausdorff算子.它包括了许多著名的算子作为其特殊情形.我们得到了这些算子在加权Morrey-Herz空间上有界的充分必要条件,还得到了相应的新的算子范数不等式.它们是许多著名结果有意义的改进和推广.     

Hilbert空间H中带符号广义框架的一个刻画

本文研究了可分的Hilbert空间H中带符号广义框架,利用算子理论方法,给出了H中一族向量{hm}m∈M是一个带符号广义框架当且仅当带符号广义框架的框架算子的正部S 和负部S-是有界线性算子,讨论了H中带符号广义框架的框架算子S的可逆性,并且得到了H中每个向量f关于带符号广义框架{hm}m∈M和其对偶带符号广义框架{~hm}m∈M的表示式.     

On sum and stability of g-frames in Hilbert spaces

In this paper, we give some new results on sum and stability of g-frames in Hilbert spaces. Since the finite sum of g-frames may not be a g-frame for the Hilbert space, we give a necessary and sufficient condition and some sufficient conditions for the finite sum of g-frames to be a g-frame. We also show that every g-sequence in Hilbert space can be expanded to a tight g-frame by adding a linear bounded operator. Moreover, we obtain some sufficient conditions under which g-frames (and the finite sum of g-frames) are stable under small perturbations.     

Banach空间中基于Neumann引理的拟线性广义逆扰动定理

本文给出Banach空间中闭线性算子的Moore-Penrose有界拟线性投影广义逆的一种新的扰动分析方法.运用的核心工具是广义Neumann引理,这与以往其他结果中所运用的广义Banach引理的处理方法极为不同,得到了闭线性算子的MoorePenrose有界拟线性广义逆的又一个扰动定理及三个误差界不等式.     

广义框架和广义Riesz基的摄动

引入并研究了Banach空间X中的Bessel集、广义框架与广义Riesz基．对X中的任一Bessel集{gm}m∈M，定义有界线性算子T：L^2（P）→X^＊，利用算子丁，给出了Bessel集与广义框架的等价刻画．同时讨论了广义框架和广义Riesz基的摄动．     

Banach空间上p-fusion框架的若干等价描述

本文说明Banach空间上p-fusion框架和p-框架有紧密联系.应用分析算子和合成算子给出p-fusion Bessel序列、p-fusion框架和q-fusion Riesz基的等价描述.     

Duality in reconstruction systems

We consider reconstruction systems (RS’s), which are G-frames in a finite dimensional setting, and that includes the fusion frames as projective RS’s. We describe the spectral picture of the set of RS operators for the projective systems with fixed weights. We also introduce a functional defined on dual pairs of RS’s, called the joint potential, and study the structure of the minimizers of this functional. In the case of irreducible RS’s the minimizers are characterize as the tight systems. In the general case we give spectral and geometric characterizations of the minimizers of the joint potential. At the end of the paper we show several examples that illustrate our results.     

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.     

K-g-框架及其对偶

本文主要探讨K-g-框架及其对偶.首先讨论K-g-框架和g-框架的关系,然后给出一些充分条件,使得在此条件下K-g-框架与g-Bessel序列经过有界线性算子或非零复有界序列作用后的和仍然是K-g-框架.此外,又给出K-g-框架求和的两种特殊形式.最后,研究K-g-框架在闭子空间R(K)上的对偶,以及利用近似对偶构建K...     

Hilbert空间中的g-Riesz框架

g-框架作为Hilbert空间中的推广框架最近被提出,它们有许多和框架类似的性质,但并不是所有的性质都是相似的.Christensen已指出了每个Riesz框架都包含一个Riesz基.本文指出并不是所有的g-Riesz框架都包含一个g-Riesz基,但我们得到了每个g-Riesz框架都包含一个无冗g-框架,同时给出了H...     

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.     

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.     

Invertibility of the Difference of Idempotents

We study conditions equivalent to the invertibility of f -g when f and g are idempotents in a unital ring, and give applications to bounded linear operators in Banach and Hilbert spaces. In the setting of rings we are able to show that many conditions previously linked to finite dimensionality, rank equalities, norm topology of bounded linear operators or to properties of C *-algebras can be in fact proved by simple algebraic arguments.     

Invertible sequences of bounded linear operators

In this paper, we study the invertibility of sequences consisting of finitely many bounded linear operators from a Hilbert space to others. We show that a sequence of operators is left invertible if and only if it is a g-frame. Therefore, our result connects the invertibility of operator sequences with frame theory.