Frames, Riesz bases, Hilbert C*-modules
Since the discovery in the early 1950's, frames have emerged as an important tool in signal processing, image processing, data compression and sampling theory etc. Today, powerful tools from operator theory and Banach space theory are being introduced to the study of frames producing deep results in frame theory. In recent years, many mathematicians generalized the frame theory from Hilbert spaces to Hilbert C*-modules and got significant results which enrich the theory of frames. Also there is growing evidence that Hilbert C*-modules theory and the theory of wavelets and frames are tightly related to each other in many aspects. Both research fields can benefit from achievements of the other field. Our purpose of this dissertation is to work on several basic problems on frames for Hilbert C*-modules. We first give a very useful characterization of modular frames which is easy to be applied. Using this characterization we investigate the modular frames from the operator theory point of view. A condition under which the removal of element from a frame in Hilbert C*-modules leaves a frame or a non-frame set is also given. In contrast to the Hilbert space situation, Riesz bases of Hilbert C*-modules may possess infinitely many alternative duals due to the existence of zero-divisors and not every dual of a Riesz basis is again a Riesz basis. We will present several such examples showing that the duals of Riesz bases in Hilbert $C^*$-modules are much different and more complicated than the Hilbert space cases. A complete characterization of all the dual sequences for a Riesz basis, and a necessary and sufficient condition for a dual sequence of a Riesz basis to be a Riesz basis are also given. In the case that the underlying C*-algebra is a commutative W*-algebra, we prove that the set of the Parseval frame generators for a unitary group can be parameterized by the set of all the unitary operators in the double commutant of the unitary group. Similar result holds for the set of all the general frame generators where the unitary operators are replaced by invertible and adjointable operators. Consequently, the set of all the Parseval frame generators is path-connected. We also prove the existence and uniqueness of the best Parseval multi-frame approximations for multi-frame generators of unitary groups on Hilbert C*-modules when the underlying C*-algebra is commutative. For the dilation results of frames we show that a complete Parseval frame vector for a unitary group on Hilbert C*-module can be dilated to a complete wandering vector. For any dual frame pair in Hilbert C*-modules, we prove that the pair are orthogonal compressions of a Riesz basis and its canonical dual basis for some larger Hilbert C*-module. For the perturbation of frames and Riesz bases in Hilbert C*-modules we prove that the Casazza-Christensen general perturbation theorem for frames in Hilbert spaces remains valid in Hilbert C*-modules. In the Hilbert space setting, under the same perturbation condition, the perturbation of any Riesz basis remains a Riesz basis. However, this no longer holds for Riesz bases in Hilbert C*-modules. We also give a complete characterization on all the Riesz bases for Hilbert C*-modules such that the perturbation (under Casazza-Christensen's perturbation condition) of a Riesz basis still remains a Riesz basis.
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Doctor of Philosophy (Ph.D.)
College of Sciences
Length of Campus-only Access
Doctoral Dissertation (Open Access)
Jing, Wu, "Frames In Hilbert C*-modules" (2006). Electronic Theses and Dissertations. 899.