Research On Frames And Some Generalized Frames In Hilbert C~*-modules

Frames, emerged as an important tool, have been extensively applied in image and signal processing, data compression, sampling theory, system modelling, code and communication etc, since the introduction in1950’s. Today, powerful tools from operator theory and Banach space theory are being introduced to the study of frames producing deep results in frame theory.In2002, Frank and Larson defined the standard frames in Hilbert C*-modules. And then g-frames, fusion frames and continuous g-frames were also generalized from Hilbert spaces to Hilbert C*-modules by some mathematicians. They got some significant results which enrich the theory of frames. It is well-known that the theory of Hilbert C*-modules is quite different from that of Hilbert spaces, so the generalizations of frame theory from Hilbert spaces to Hilbert C*-modules are not trivial. And 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. This makes the study of frames in Hilbert C*-modules important and interesting.The purpose of this dissertation is to work on several basic problems on frames, g-frames, continuous g-frames and fusion frames in Hilbert C*-modules. We also define two new frames in Hilbert C*-modules. Our main work is spread across five chapters. Let’s describe it concretely.In Chapter Three, we study some properties of frames in Hilbert C*-modules. A completed proof for a known result is presented by means of the Moore-Penrose inverses of adjointable operators. We discuss the applications of adjointable operators in the construction of frames and the transforms of frames and dual frames. We obtain a sufficient and necessary condition for a modular frame to be redundant and several conditions under which the removal of some elements from a frame in Hilbert C*-modules leaves a frame are also given. Moreover, we obtain new results for the perturbation of frames and Riesz bases in Hilbert C*-modules.In Chapter Four, we discuss some equalities and inequalities for g-frames and the duality of g-frames in Hilbert C*-modules. We obtain some new equalities for g-frames which cover some existing results and, especially, some new types of inequalities for g-frames are given. We define the approximately dual and pseudo-dual for g-frames in Hilbert C*-modules and the concept of dual g-frames determined by invertible operators is introduced. We get sufficient and necessary conditions for two g-Bessel sequences to be pseudo-dual g-frames and dual g-frames determined by invertible operators. We also give some proper conditions for two g-Bessel sequences to be approximately dual g-frames and the perturbation of dual g-frames determined by invertible operators is discussed. The concepts of g-frame systems and Q-dual g-frames are introduced for the purpose of constructing and studying the relations between two g-frames with respect to different sequences of closed submodules and an equivalent condition for two g-frame systems to be Q-dual g-frame systems is obtained.In Chapter Five, we give a characterization on the existence of continuous g-frames in Hilbert C*-modules. We improve one existing result by introducing the so-called modular continuous g-Riesz basis. We give two sufficient and necessary conditions under which two continuous g-frames are similar and in particular, a perturbation result for alternate dual continuous g-frames is obtained by observing the difference between a canonical dual and an alternate dual.In Chapter Six, we improve the original definition for fusion frames in Hilbert C*-modules. Characterizations of Bessel fusion sequences and fusion frames are given and the perturbation of fusion frames is discussed. We also study the minimality of fusion frame coefficients and a condition for one fusion frame be a non-frame set after deleting one vector from the fusion frame is given.In Chapter Seven, we define two new frames in Hilbert C*-modules, that is, frames with A-valued bounds and adjointable-operators frames. We show that frames with A-valued bounds are equivalent to g-frames in Hilbert C*-modules and we prove that Hilbert C*-module frames, g-frames, continuous g-frames, fusion frames and frames with A-valued bounds are all adjointable-operators frames.