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.     

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.     

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).     

Similarity and parameterizations of dilations of pairs of dual group frames in Hilbert spaces

In this paper, firstly, in order to establish our main techniques we give a direct proof for the existence of the dilations for pairs of dual group frames. Then we focus on proving the uniqueness of such dilations in certain sense of similarity and giving an operator parameterization of the dilations of all pairs of dual group frames for a given group frame. We show that the operators which transform different dilations are of special structured lower triangular.     

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.     

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.     

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.     

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-frames and g-Riesz bases

G-frames are generalized frames which include ordinary frames, bounded invertible linear operators, as well as many recent generalizations of frames, e.g., bounded quasi-projectors and frames of subspaces. G-frames are natural generalizations of frames and provide more choices on analyzing functions from frame expansion coefficients. We give characterizations of g-frames and prove that g-frames share many useful properties with frames. We also give a generalized version of Riesz bases and orthonormal bases. As an application, we get atomic resolutions for bounded linear operators.     

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.     

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.     

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.     

Partially isometric dilations of noncommuting -tuples of operators

Given a row contraction of operators on a Hilbert space and a family of projections on the space that stabilizes the operators, we show there is a unique minimal joint dilation to a row contraction of partial isometries that satisfy natural relations. For a fixed row contraction the set of all dilations forms a partially ordered set with a largest and smallest element. A key technical device in our analysis is a connection with directed graphs. We use a Wold decomposition for partial isometries to describe the models for these dilations, and we discuss how the basic properties of a dilation depend on the row contraction.

     

An Invertibility Criterion in a C*-Algebra Acting on the Hardy Space with Applications to Composition Operators

In this paper, we prove an invertibility criterion for certain operators which is given as a linear algebraic combination of Toeplitz operators and Fourier multipliers acting on the Hardy space of the unit disc. Very similar to the case of Toeplitz operators, we prove that such operators are invertible if and only if they are Fredholm and their Fredholm index is zero. As an application, we prove that for “quasi-parabolic” composition operators the spectra and the essential spectra are equal.     

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.     

Left and right generalized Drazin invertible operators

In this paper, we define and study the left and the right generalized Drazin inverse of bounded operators in a Banach space. We show that the left (resp. the right) generalized Drazin inverse is a sum of a left invertible (resp. a right invertible) operator and a quasi-nilpotent one. In particular, we define the left and the right generalized Drazin spectra of a bounded operator and also show that these sets are compact in the complex plane and invariant under additive commuting quasi-nilpotent perturbations. Furthermore, we prove that a bounded operator is left generalized Drazin invertible if and only if its adjoint is right generalized Drazin invertible. An equivalent definition of the pseudo-Fredholm operators in terms of the left generalized Drazin invertible operators is also given. Our obtained results are used to investigate some relationships between the left and right generalized Drazin spectra and other spectra founded in Fredholm theory.     

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.).     

SOME RESULTS ON CONTROLLED FRAMES IN HILBERT SPACES

We use two appropriate bounded invertible operators to define a controlled frame with optimal frame bounds.We characterize those operators that produces Parseval controlled frames also we state a way to construct nearly Parseval controlled frames.We introduce a new perturbation of controlled frames to obtain new frames from a given one.Also we reduce the distance of frames by appropriate operators and produce nearly dual frames from two given frames which are not dual frames for each other.     

Constructing pairs of dual bandlimited framelets with desired time localization

For sufficiently small translation parameters, we prove that any bandlimited function ψ, for which the dilations of its Fourier transform form a partition of unity, generates a wavelet frame with a dual frame also having the wavelet structure. This dual frame is generated by a finite linear combination of dilations of ψ with explicitly given coefficients. The result allows a simple construction procedure for pairs of dual wavelet frames whose generators have compact support in the Fourier domain and desired time localization. The construction is based on characterizing equations for dual wavelet frames and relies on a technical condition. We exhibit a general class of function satisfying this condition; in particular, we construct piecewise polynomial functions satisfying the condition.