Hilbert空间中的斜对偶g-框架

在Hilbert空间中把斜对偶原理推广到更一般的g-框架.我们给出了{A_j:j∈J}是g-框架{F_j:j∈J}的一个斜对偶g-框架的等价条件,还给出了一个斜对偶g-框架对是对称的充分条件.最后,在不同的条件下构造了几对斜对偶g-框架.     

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.     

Characterizations of g-frames and g-Riesz bases in Hilbert spaces

In this paper, we introduce the pre-frame operator Q for the g-frame in a complex Hilbert space, which will play a key role in studying g-frames and g-Riesz bases etc. Using the pre-frame operator Q, we give some necessary and sufficient conditions for a g-Bessel sequence, a g-frame, and a g-Riesz basis in a complex Hilbert space, which have properties similar to those of the Bessel sequence, frame, and Riesz basis respectively. We also obtain the relation between a g-frame and a g-Riesz basis, and the relation of bounds between a g-frame and a g-Riesz basis. Lastly, we consider the stability of a g-frame or a g-Riesz basis for a Hilbert space under perturbation.     

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).     

G-frames and g-frame sequences in Hilbert spaces

In this paper, we first determine the relations among the best bounds A and B of the g-frame, the g-frame operator S and the pre-frame operator Q and give a necessary and sufficient condition for a g-frame with bounds A and B in a complex Hilbert space. We also introduce the definition of a g-frame sequence and obtain a necessary and sufficient condition for a g-frame sequence with bounds A and B in a complex Hilbert space. Lastly, we consider the stability of a g-frame sequence for a complex Hilbert space under perturbation.     

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.     

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.     

Hilbert C~*-模上的广义g-框架

本文主要研究了Hilbert C~*-模上广义g-框架.给出了Hilbert C~*-模上的广义g-框架,广义g-框架算子,广义典则对偶g-框架及广义交错对偶g-框架的定义,并证明了有关广义g-框架的一些结果.     

g-Besselian frames in Hilbert spaces

In this paper, we introduce the concept of a g-Besselian frame in a Hilbert space and discuss the relations between a g-Besselian frame and a Besselian frame. We also give some characterizations of g-Besselian frames. In the end of this paper, we discuss the stability of g-Besselian frames. Our results show that the relations and the characterizations between a g-Besselian frame and a Besselian frame are different from the corresponding results of g-frames and frames.     

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.     

Excess of a class of g-frames

In this paper, we study the relationship between frames for the super Hilbert space H⊕H and g-frames for H with respect to C². We show that a g-frame associated with a frame for H⊕H remains a g-frame whenever any one of its elements is removed. Furthermore, we show that the excess of such a g-frame is at least dimH.     

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.).     

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...     

Decompositions of a Krein space in regular subspaces invariant under a uniformly bounded C0-semigroup of bi-contractions

We give necessary and sufficient conditions under which a C₀-semigroup of bi-contractions on a Krein space is similar to a semigroup of contractions on a Hilbert space. Under these and additional conditions we obtain direct sum decompositions of the Krein space into invariant regular subspaces and we describe the behavior of the semigroup on each of these summands. In the last section we give sufficient conditions for the co-generator of the semigroup to be power bounded.     

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.     

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.     

Hilbert C~*-模中g-框架的一些性质

本文运用算子理论方法,给出Hilbert C~*-模中g-框架的一些性质并讨论g-框架的扰动性,得到g-框架的和的一些刻画,所得结果推广和改进了已有的结果.     

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.     

Norm inequalities for sums of two basic elementary operators

We use certain norm inequalities for 2×2 operator matrices to establish norm inequalities for sums of two basic elementary operators on a Hilbert space. Further, we give necessary and sufficient conditions under which the norm of the above sum of elementary operators attains its optimal value. Applications of the inequalities obtained are also considered.