Perturbation Of Frames And Riesz Bases For Hilbert Spaces And Their Orthogonal Decomposition

Frames for a Hilbert space H were formally defined by R. J. Duffin and A. G. Schaeffer in 1952. Frames play an important role in the development of wavelet analysis. Frames for a Hilbert space H can be viewed as generalized bases for H. This is the value of frames. Functional analysis is an ancient branch of mathematics. In this thesis, we give some results about the perturbation of frames and Riesz basis, and some results about their decomposition.This thesis consists of four chapters.In chapter 1, we introduce some concepts such as Bessel sequence, frame, tight frame, dual frame, independent frame, Riesz basis, pre-frame operator, frame operator and discuss some basic properties of them. We also give some examples. We build the correspondence between frames and operators. A sequence of elements {fi}i I in # is a frame for H if and only if there is an operator Tf : H -- l2(I) such that Tfx = {(x, fi)}i I l2(I), and Tf is bounded below if and only if there is an operator T : H H so that Tei = fi, i I, and T is an onto map. {fi}i 1 is a Riesz basis if and only if the operator T is invertible..In chapter 2, since all the frames for H form an open subset in H, we can obtain a frame by translating, modulating another frame. Firstly, we give some sufficient conditions for the sequences to be frames for H when the sequence {fi}i I is a frame for H. Secondly, suppose that {fi}i I is a frame for H, {gi}i I is a sequence of elements in H, we use the dual of the pre-frame operator to give some sufficient conditions for {gi}i I to be a frame for H. Finally, suppose that f = {fj}i I is a frame and g = {gi}i I is a Bessel sequence for H, we give some sufficient conditions for f g to be frames by pre-frame operators.In chapter 3, suppose that {fi}i I is a Riesz basis for H(span{fi}i i, {gi}i I is a Bessel sequence for H, we give some sufficient conditions for {gi}i I to be a Riesz Basis for H (span{gi}i I). Invertible operators have the preservation stability under perturbations. We applied this property to the perturbation of Riesz bases.In chapter 4, motivated by the correspondence between frames (Riesz bases) and onto maps (invertible maps). Use the polar decomposition theorem of operators, we obtain the result that every frame is a sum of three (but not two ) orthogonalbases. And a frame for H can be written as a linear combination of two orthogonal bases for H if and only if it is a Riesz basis for H.