Sub Mbekhta Space And Complex Analysis

The spectral theory of bounded linear operaors is an important direction in functional analysis. In general, people study the spectrum of an operator by means of the kernel and range of A ( N(A) and R(A) ). In [8],M.Mekhta generalized the concept of kernel and range in 1987, he introduced twe important subspaces Ho(A) and K(A). In[6],Prof Gong and Wang Libin discussed the property of Ho(A) and K(A) when A is an bounded linear operator, he also study the spectral theory of compact operator with these twe subspaces. Motivated by [6] , we discuss the property of Ho(A) and K(A) continuously and and study the spectral theory of normal operators and compact normal operators by means of Ho(A) and K(A) in this paper.This paper is composed of four parts, The main results are the following: Charpter 1 Preface Charpter 2 The Property of H0(A) and K(A)This charpter discusses the property of Ho(A) and K(A] when A is an bounded linear operator and comes to a series of conclusions. The main results: are isolated points in thenCor2.4 is an isolated point in . If satisfies ,then a (A) has no more than two isolated points .at this time we have the following results:(l has only n different isolated points;Prop2.7 Let (X). If 0 is a isolated point in then A is completely reduced by the pair (Ho(A),K(A)) andCharpter 3 The Spectrum of Normal OperatorThis charpter discusses the spectral theory of normal operators by means of Ho(A) and K(A), and discusses the relationsof Ho(A) and N(A) and that of R(A) and K(A) when A is a normal operator, this part also studies the new properties of normal operator, The main results are the following:Theorem3. 9 L(H) be a normal operator and , for every If is an isolated point in .Prop3.15 Let A be a normal operator. If there exists an isolated point Aoin a (A), then there exists a normal operator , we have:(1) AB = BA;(2) A*B = BA*,AB* = B'A\(3) B,B* are completely reduced by the pair Charpter 3 The Spectrum of Normal OperatorThis charpter discusses the spectral theory of compact normal operators by means of H0(A) and K(A), The main results are the following:Theorem4.4 Let A be a compact normal operator. Then the following state-ments are equivalent:(1)0 is an isolated point in (2) A is a finite rank operator; is the spectral projector associated with .Theorem4.5 Let A be a compact normal operator, is an isolated point in Prop4.13 Let A be a compact normal operator, AO is an isolated point in a(A). Then(1) There exist controvertible normal operators E, F, such thatand 0 is an isolated point in is an isolated point in ,(2) There exist a series of operator An, such that...