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.     

Isometries on spaces of vector valued Lipschitz functions

This paper gives a characterization of a class of surjective isometries on spaces of Lipschitz functions with values in a finite dimensional complex Hilbert space.     

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...     

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.     

Infinite-Dimensional Measure Spaces and Frame Analysis

We study certain infinite-dimensional probability measures in connection with frame analysis. Earlier work on frame-measures has so far focused on the case of finite-dimensional frames. We point out that there are good reasons for a sharp distinction between stochastic analysis involving frames in finite vs. infinite dimensions. For the case of infinite-dimensional Hilbert space ?, we study three cases of measures. We first show that, for ? infinite dimensional, one must resort to infinite dimensional measure spaces which properly contain ?. The three cases we consider are: (i) Gaussian frame measures, (ii) Markov path-space measures, and (iii) determinantal measures.     

Austere and arid properties for PF submanifolds in Hilbert spaces

Austere submanifolds and arid submanifolds constitute respectively two different classes of minimal submanifolds in finite dimensional Riemannian manifolds. In this paper we introduce the concepts of these submanifolds into a class of proper Fredholm (PF) submanifolds in Hilbert spaces, discuss their relation and show examples of infinite dimensional austere PF submanifolds and arid PF submanifolds in Hilbert spaces. We also mention a classification problem of minimal orbits in hyperpolar PF actions on Hilbert spaces.     

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.     

Inertia theorems for operator pencils and applications

In this paper we obtain general infinite dimensional inertia theorems for linear pencils in Hilbert space which cover previously known results for the finite dimensional case and for block weighted shifts. Connections with definite subspaces for contractions in spaces with indefinite metric are discussed.     

Characterization of the closedness of the sum of two shift-invariant spaces

We first present a formula for the supremum cosine angle between two closed subspaces of a separable Hilbert space under the assumption that the ‘generators’ form frames for the subspaces. We then characterize the conditions that the sum of two, not necessarily finitely generated, shift-invariant subspaces of L²(Rd) be closed. If the fibers of the generating sets of the shift-invariant subspaces form frames for the fiber spaces a.e., which is satisfied if the shift-invariant subspaces are finitely generated or if the shifts of the generating sets form frames for the respective subspaces, then the characterization is given in terms of the norms of possibly infinite matrices. In particular, if the shift-invariant subspaces are finitely generated, then the characterization is given wholly in terms of the norms of finite matrices.     

Mixed-type reverse order law for products of three operators

We present new results related to the mixed-type reverse order law for the Moore-Penrose inverse of various products of three operators on Hilbert spaces. Some finite dimensional results are extended to infinite dimensional settings.     

THE FUNCTIONAL DIMENSION OF SOME CLASSES OF SPACES

The functional dimension of countable Hilbert spaces has been discussed by some authors. They showed that every countable Hilbert space with finite functional dimension is nuclear. In this paper the authors do further research on the functional dimension, and obtain the following results: (1) They construct a countable Hilbert space, which is nuclear, but its functional dimension is infinite. (2) The functional dimension of a Banach space is finite if and only if this space is finite dimensional. (3) Let B be a Banach space, B* be its dual, and denote the weak * topology of B* by σ(B*,B). Then the functional dimension of (B*,σ(B*,B)) is 1. By the third result, a class of topological linear spaces with finite functional dimension is presented.     

Gabor Analysis on Finite Product of Cyclic Groups

We present a complete characterization of Gabor frame operators on finite dimensional Hilbert space $L^2(\Gamma)$, where $\Gamma = \mathbb{Z}_{m_1} \times \mathbb{Z}_{m_2} \times \dots \times \mathbb{Z}_{m_p}$ and $m_1,m_2,\dots,m_p$ are positive integers. The notion of generalized $B$-modulation and generalized $B$-translation is introduced and some significant properties of the generalized pseudo $B$-Gabor like frames are discussed.     

Some Properties of GMRES in Hilbert Spaces

Our purpose in this work is to explore the properties of GMRES in Hilbert spaces. We extend to the infinite dimensional context some main results that are known to hold in the finite dimensional case. A key assumption for these extensions is that the involved linear operator is an algebraic operator.     

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.     

Fusion frames and <Emphasis Type="Italic">G</Emphasis>-frames in Banach spaces

Fusion frames and g-frames in Hilbert spaces are generalizations of frames, and frames were extended to Banach spaces. In this article we introduce fusion frames, g-frames, Banach g-frames in Banach spaces and we show that they share many useful properties with their corresponding notions in Hilbert spaces. We also show that g-frames, fusion frames and Banach g-frames are stable under small perturbations and invertible operators.     

A remark on the Brézis–Browder theorem in finite dimensional spaces

Brézis and Browder gave an important characterization of maximal monotonicity for linear operators, in reflexive Banach spaces. In this paper, we show that this characterization also preserves the maximal p-monotonicity, in finite dimensional spaces.     

Existence of equilibria of set-valued maps on bounded epi-Lipschitz domains in Hilbert spaces without invariance conditions

In this paper, we provide a new result of the existence of equilibria for set-valued maps on bounded closed subsets K of Hilbert spaces. We do not impose either convexity or compactness assumptions on K but we assume that K has epi-Lipschitz sections, i.e. its intersection with suitable finite dimensional spaces is locally the epigraph of Lipschitz functions. In finite dimensional spaces, the famous Brouwer theorem asserts the existence of a fixed point for a continuous function from a compact convex set K to itself. Our result could be viewed as a kind of generalization of this classical result in the context of Hilbert spaces and when the function (or the set-valued map) does not necessarily map K into itself (K is not invariant under the map). Our approach is based firstly on degree theory for compact and for condensing set-valued maps and secondly on flows generated by trajectories of differential inclusions.     

Reverse order law for generalized inverses of multiple operator product

In this paper, we consider the reverse order law for generalized inverses of operators on Hilbert spaces. We derive necessary and sufficient conditions for various inclusions concerning the reverse order law for generalized inverses of multiple operator product. We extend the finite dimensional results from (Wei M. Reverse order laws for generalized inverses of multiple matrix products. Linear Algebra Appl. 1999;293:13.) to infinite dimensional settings.