Redundant Frames And Some Problems In C～*- Algebras

In this paper, some concepts about redundant frames are introduced. Necessary and sufficient conditions for a frame to be redundant are given. Especially, it is proved that frame is nonredundant if and only if it is a Riesz basis. On the basis of this, effects on the wavelet reconstruction with redundant frames are discussed.In a unital C^*-algebra A, the inverse problem of contraharmonic and geometric means of positive invertible elements in A is discussed. On the other hand, we define idempotent operator algebra acting on a Hilbert space H. Some important properties and results of idempotent operator algebras are given. This paper is divided into four chapters.In Chapter 1, some notions and definitions are introduced and some well-known theorems are given. In Section 1. 1, we give some technologies and notations, and introduced the definitions of redundant frames, nonredundant frame in Hilbert space, invertible positive operator, frame operator, controharmonic and geometric means, idempotent opertor algebras and so on. In Section 1. 2, we given some well-known theorems for the convenience of the following statements.In Chapter 2, we first introduce the concept of redundant frames and give a necessary and sufficient conditions for a frame to be redundant. Some equivalent characterization of redundant frames are given.In Chapter 3, we study two nonlinear equations and in the system(1) is positively solvable if and only if a^-1／2ba^-1／2≤1. In this case (t, y_±) defined asy₊=1／2(t+t#(t-4bt^-1b))=1／2(t+(t+2b)#(t-2b)), (3)y_-=1／2#(t-4bt^-1b))=1／2(t-(t+2b)), (4)where t=1／2(a+a#(a+8ba^-1b)) are the extreme solutions in the sense that y_-≤y≤y₊ for all positive solutions(x, y) of (2).In Chapter 4, we define idempotent operator algebras acting on a Hilbert space H. We present some important properties of idempotent operator algebras. In addition, some sufficient conditions for a operator algebra to be an idempotent operator algebra are presented. Let H be a finite-dimensional Hilbert space andΩbe a subalgebra of B(H). Suppose that dim H= n. IfΩis unital, thenΩis an idempotent operator algebra. IfΩis not unital and for each element A inΩ, there exists a nonzero complex numberλ_A such that ran(A)(?)ker(A-λ_A), thenΩis an idempotent operator algebra.