Modular Frames And Continuous Frames

A frame on a Hilbert space is a natural generalization of “Riesz basis”, which has a widelyapplication in the areas of information communications etc. In recent years, many researchers arefocused on the study of frames by using the approach of operator theory and operator algebras; in fact,this represents one of the most active research areas in frame theory. Frames have several extendedversion, e.g., g-frames, modular frames, fusion frames and operator-valued frames and so on.In general, the sum of two frames is not necessarily a frame.G. Gasazza et al in2009characterizedthe properties of sums of discrete frames （or Bessel sequences） in Hilbert spaces. Motivated by this,we in this thesis will mainly study the sums of Hilbert C^*-modular frames or respectively, continu-ous frames in Hilbert spaces. Several sufficient conditions for the sum of two modular frames （or, onemodular frame and one Bessel sequence） still to be a modular frame are given, and by using the origi-nal frame bounds, the form of frame bounds of the sums is described. In the case of continuous framesin Hilbert spaces, on the one hand, we get that some interesting results similar to the case of modularframes; on the other hand, we give some sufficient conditions （or necessary and sufficient conditions）under which the vector-valued functionT₁F₁+T₂F₂becomes a frame, and characterize the frame ope-rators and frame bounds of T₁F₁+T₂F₂. Here,F₁,F₂denotes continuous frames or Bessel mappingsandT₁,T₂are bounded linear operators.In addition, let F be a continuous frame from a measure space（Ω,μ）, to a Hilbert space H. Ap-plying the analysis operatorTF, we introduce a special subspace H_F^p={x∈H::T_Fx∈Lp（Ω,μ）},1≦p＜∞, and obtain that some interesting properties of this H_F^p subspace.