Operator Frames for Banach Spaces

In this paper, operator Bessel sequences, operator frames, Banach operator frames, Operator Riesz bases for Banach spaces and dual frames of an operator frame are introduced and discussed. The necessary and sufficient condition for a Banach space to have an operator frame, a Banach operator frame or an operator Riesz basis are given. In addition, operator frames and operator Riesz bases are characterized by the analysis operator of operator Bessel sequences.     

Frames in generalized Fock spaces

Let A denote a real linear transformation on Cn which is symmetric and positive-definite relative to the real inner product Re〈z,w〉, z,w∈Cn. Let FA(Cn) denote the Fock space consisting of holomorphic functions on Cn which are square integrable with respect to the Gaussian measure . For w∈Cn, let , z∈Cn, where KA is the reproducing kernel for FA(Cn). The main aim of this paper is to show that there exist a,b>0 such that the set of functions forms a frame in FA.     

Proximinality in Banach spaces

In this paper, we study approximatively τ-compact and τ-strongly Chebyshev sets, where τ is the norm or the weak topology. We show that the metric projection onto τ-strongly Chebyshev sets are norm-τ continuous. We characterize approximatively τ-compact and τ-strongly Chebyshev hyperplanes and use them to characterize factor reflexive proximinal subspaces in τ-almost locally uniformly rotund spaces. We also prove some stability results on approximatively τ-compact and τ-strongly Chebyshev subspaces.     

Bimeasures in Banach spaces

In this paper we study integration with respect to a bimeasure with finite semivariation. The bimeasures as well as the functions to be integrated, take on their values in Banach spaces. Entrata in Redazione il 25 settembre 1998 e, in versione riveduta, il 25 giugno 1999.     

Banach空间中框架的对偶原理

Gabor理论中的对偶原理(例如Ron-Shen对偶原理和Wexler-Raz双正交关系)在研究Gabor系统时起到了至关重要的作用. 对Banach空间中的任意序列, 该文定义了仅依赖两组 p-Riesz基的一个相关的序列(Riesz -对偶序列), 研究它与前一组序列相关的性质. 推广了P. G. Gasazza、G. Kutyniok和M. C. Lammers在可分Hilbert空间中框架的对偶原理的一些结果.     

Banach空间上的框架与拟Riesz基

该文首先给出Ｂａｎａｃｈ空间上的框架与拟Ｒｉｅｓｚ基的充要条件,其次讨论Ｂａｎａｃｈ空间上的框架和拟Ｒｉｅｓｚ基的稳定性,特别地,讨论在Ｂａｎａｃｈ空间上关于框架与拟Ｒｉｅｓｚ基的广义Ｐａｌｅｙ Ｗｉｅｎｅｒ定理．     

Frames for vector spaces and affine spaces

A finite frame for a finite dimensional Hilbert space is simply a spanning sequence. We show that the linear functionals given by the dual frame vectors do not depend on the inner product, and thus it is possible to extend the frame expansion (and other elements of frame theory) to any finite spanning sequence for a vector space. The corresponding coordinate functionals generalise the dual basis (the case when the vectors are linearly independent), and are characterised by the fact that the associated Gramian matrix is an orthogonal projection. Existing generalisations of the frame expansion to Banach spaces involve an analogue of the frame bounds and frame operator.The potential applications of our results are considerable. Whenever there is a natural spanning set for a vector space, computations can be done directly with it, in an efficient and stable way. We illustrate this with a diverse range of examples, including multivariate spline spaces, generalised barycentric coordinates, and vector spaces over the rationals, such as the cyclotomic fields.     

Banach空间上的框架与Riesz基

本文讨论Banach空间上框架、无冗框架与Riesz基之间的关系及它们的稳定性．     

Ergodic Banach spaces

We show that any Banach space contains a continuum of non-isomorphic subspaces or a minimal subspace. We define an ergodic Banach space X as a space such that E₀ Borel reduces to isomorphism on the set of subspaces of X, and show that every Banach space is either ergodic or contains a subspace with an unconditional basis which is complementably universal for the family of its block-subspaces. We also use our methods to get uniformity results. We show that an unconditional basis of a Banach space, of which every block-subspace is complemented, must be asymptotically c₀ or ?_p, and we deduce some new characterisations of the classical spaces c₀ and ?_p.     

Confined Banach spaces

We discuss smoothness of theWeyl functional calculus and use it to prove that every C*-algebra is a confined Banach space. Received: 17 August 2005     

The stability of Banach frames in Banach spaces

In this paper, we give some equivalent conditions on a Banach frame for a Banach space by using the pseudoinverse operator. We also consider the stability of a Banach frame for a Banach space X with respect to Xd or an Xd-frame for a Banach space X under perturbation. These results generalize and improve the related works of Balan, Casazza, Christensen, Stoeva and Jian et al.     

Wold decomposition in Banach spaces

We propose a natural analog of the Wold decomposition in the case of a linear noninvertible isometry V in a Banach space X. We obtain a criterion for the existence of such a decomposition. In a reflective space, this criterion is reduced to the existence of the linear projection P: X → VX with unit norm. Separately, we discuss the problem of the Wold decomposition for the isometry V _φ induced by an epimorphism φ of a compact set H in the space of continuous functions C(H). We present a detailed study of the mapping z → z ^m of the circle |z| = 1 with an integer m ≥ 2.