Generalized Frames In Hilbert W~*-Modules

This thesis is devoted to generalizing the notion of frame to the Hilbert W*-module setting by combining the definitions of discrete frame in Hilbert C*-modules and generalized frame in Hilbert spaces, and obtain the notions of generalized frame, generalized tight frame, generalized normalized tight frame and generalized dual frame in Hilbert W*-modules. In this thesis, the standard generalized frames in Hilbert W*-modules are primarily studied, also some definitions such as analysis operator, frame operator and dual frame for a given generalized frame are introduced, and the reconstruction formula which is very important in Hilbert C*-modules and Hilbert spaces is available. The definitions of strong disjointness, disjointness and weak disjointness and their equivalent conditions which are similar to the case in Hilbert spaces are given in chapter 2, while the proofs which we mainly use operator-theoretic-methods are more complicated. Some important results are showed, such as under some other assumptions, the sum or the direct sum of two or more generalized frames also form a generalized frame under the assumptions of strong disjointness, disjointness or weak disjointness, respectively. It is known that the most important problem of frames is perturbation, in this thesis, some perturbation conditions about generalized frames, accompanying with the new bounds under perturbations and some other useful results, are intensively investigated. At the end of this thesis, the equalities and inequalities about Hilbert W*-modular frames, especially the generalized Parseval frame equality are studied.