Two Combinations Of Unitary Operators And Representations Or Perturbation Of Frames

In this paper, we discuss the two combinations of unitary operators and the perturbation of closed range operators in B(H). Then we apply these results to frame theory , and we get a series of new results about the representations and perturbation of frames.This thesis consists of four chapters.In chapter 1, we introduce some concepts such as Bessel sequence, frame, Riesz basis, and so on, then we discuss some basic properties of them. We reveal the connections between frames and operators.In chapter 2, we study and discuss the two combinations of unitary operators, and from the operator theory point of view, we do some discussions about the representations of frames. In the first section, we characterize the two combinations of unitary operators, the main results are:(l) A ∈U1 if and only if dimN(A) = dimN(A*); (2) A ∈g(H) if and only if there exist λ1,λ2 ∈ C which satisfying |λ1| ≠|λ2| and U1,U2 ∈ U(H) such that A = λ1U1 + λ2U2; (3) The set U1 is equal to the set 1/2; (4) There are three equal conditions as follows: A ∈ dg(H) if and only if dimN(A) = dimN(A*) or R(A) is not closed if and only if A ∈ dU1\. In the second section, we apply these results which we get in section one to frame theory. We obtain the result that every Bessel sequence is a sum of three (but not two) orthogonal bases or can be written as a multiple of the sum of a Riesz and an orthonormal basis. 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 Reisz basis for H.In chapter 3, we study the two frame approximation of T x. In the first section, we discuss the linear frame approximation of T x. We construct a sequence of operators (0n}neJVi such that for every x 6 H, we have x ->T^x when n ->oo. In the second section, we discuss the nonlinear iterative frame approximation of T*x. We construct a sequence {gjigjv. such that for every fixed x E H and e > 0, we have \\T^x - ]Cfe=o 9k\\ ^ jpnr||rt||. And in the final part of every section we discuss the frame approximation of frame operator S~lx.In chapter 4, we study the perturbation of closed range operators and apply the results to frame theory. In the first section, we discuss the perturbation of closed range operators.The main results are:(1) T ∈ BC(H), S is some linear operator on H, if there exist two numbers AI < 1, A2 < 1, such that for each x e H we have ||Tx- Sx|| < λ1||Tx|| + λ2||Sx||, then 5 6 BC(H); (2) Under the same assumptions as in (1), if in addition T is surjective, then so is S; (3) Under the same assumptions as in (1), if in addition T is invertible, then so is 5; (4) T ∈ B( H)and T is bounded below, S is some linear operator on ft, then S∈B(H)and S is bounded below if and only if there exists M > 0, such that for each x € ft we have ||Tx - Sx||2 < Mmin{||Tx||2, ||Sa;||2}. In the second section, we discuss the perturbation of frames, we get:(l) If {fi}i ∈ N is a Bessel sequence in ft with upper bound B, {gi}i∈N is some sequence in ft, if there exist two numbers λ1, λ2 ∈ (-1, l),such that for any finite sequence {ci}ni=1 in l2(N) we have || Σn i=1 ci(fi -gi) || < λ1|| Σni=1 cifi|| + λ2|| Σn i+1 cigi||, then {gi}i∈N is a Bessel sequence in H with upper bound B; (2) If {fi}i∈N is a frame in H with upper bound and lower bound A, B,{gi}i∈N is some sequence in H, if there exist two numbers λ1, λ2 ∈ (-1,1),such that for any finite sequence {ci}n i=1 in l2(N) we have || Σn i=1 ci(fi-gi)|| < λ1||n i=1 cifi||+λ2|| Σn i+1 cigi||. then {9i}i∈N is a frame in H with upper bound and lower bound B, A; (3) If {fi}i∈N is a frame mH,{gi}i∈N is any sequence in H, then {gi}i∈N ris a frame in H if and only if there exists a constant M > 0,such that for all x ∈ H we have Σi∈N | < x, fi-gi >2 < M mm{Σ i∈N | 2,Σi∈N|2}.