Three Types Of Frames In Banach Spaces

In this article, firstly we introduce the concepts of Xd Bessel sequences,Xd frames, Xd—Bessel sequences, Xd—frames and frames of order p in a Banach space. Then the some equivalent characterizations of Xd Bessel sequences and Xd-Bessel sequences for a Banach space X are given and some properties of Xd Bessel se-quences and Xd—Bessel sequences for a Banach space are discussed, and then we give necessary and sufficient conditions for a Banach space X to have Xd frames or Xd-frames. At the same time, we discuss the perturbation of Xd frames by the operator theory. Finally, we study the excess of frames of order p for a Banach space. This paper is organized as follows.In Chapter 1, we introduce the history and development of the theory on frames, and we give some preliminaries which will be used in following chapters.In Chapter 2, we investigate Xd Bessel sequences and Xd frames for a Banach space. It is proved that (BXXd,‖·‖) is a Banach space when Xd is a BK-space. By defining an operator Tf, an isometric isomorphism from BXXd to B(X*,Xd) is established when Xd is a BK-space and is reflexive, which provides a necessary theoretical basis for studying Xd Bessel sequences by the operator theory. Then the equivalent characterizations of Xd Bessel sequences for a Banach space are given. Moreover, it is proved that independent Xd frames and Xd Riesz bases for a Banach space X are the same. Finally, the existence and perturbation of Xd frames for a Banach space X are studyed.In Chapter 3, we investigate the properties of Xd—Bessel sequences and Xd-frames for a Banach space. It is proved that when Xd is a BK-space, Bx(Xd) is isometrically isomorphic with B(X,Xd) which implys BX(Xd) is a Banach space. When Xd is a reflexive BK-space with the basis{ei}iΛ, then some equivalent char-acterizations Xd—Bessel sequences for a Banach space X are given. Firstly, the existence of Xd—frames for a Banach space X are studyed.In Chapter 4, we study the excess of frames of order p for a Banach space. In particular, we characterize those frames for which there exist infinitely many elements that can be removed form the frame yet still leave a frame. At the same time,Realtionships with Banach space Y and lp(Y)are discussed.