Mbekhta's Subspaces And CI Operators

In [6] M.Mbekhta intoduced two importent subspaces H0(A) and K(A) in 1952. By means of Mbekhta's subspaces H0(A) and K(A),we can get better results than ever in the study of the spectral theory of bounded linear operators .If Aois the pole of R(A)of order of a , then N(P) and R(P)can be expressed in the terms of nullspace and the range of i.e.andbut we didn't find the nullspace and range played a similar role for the isolated point in a (A) At this time Mbekhta's subspaces H0(A) and K(A) fill this gap . If 0 is an isolated point of operator A thenR(P0) = H0(0 A) , N(P0) = K(0-A)andthe concept of CI opertors (consistent in invertibility with respect to multiplication) was introduced by Weibang Gong in [3] . The author also gives a complete characterization of the operator B which satisfies the spectral condition : (AB) =(BA) for every A < B(H).In this paper we discussed the applied use of Mbekhta's subspaces and the theory of CI operators . We study the spectral theory of bounded linear operators and the characterization of CI operators by way of Mbekhta's subspaces . We find a series of operators which are CI operators by the defination and the characterizationof CI operators given by Weibang Gong in [3]. We also discuss the relationship between generalized inverse of an operator and Mbekhta's subspaces and that between CI operators and generalized inverse of an operator .This thesis falls into four parts.The first part mainly explains the background and some basic knowledge which contains the sigals and concepts mentioned in this paper.In the second part we study the spectral theory of bounded linear operators.The main results are the following statements.Th 1.18 If H0(A) = H then R(A). for every Th 1.19 H0(A) if and only if (A) , for every C.Cor 1.20 If {0} H0(A) and A has SVEP atA , then A is a CI operator.Th 1.21 {0} = H0(A) and K(A) H if and only if (A)(A).Cor 1.22 If (A), then A has SVEP at A.In the third part we introduced some kinds of operators which are CI operators by the defination and characterization of CI operators . The following are the main results.Th 3.1 quasi-nilpotent operators are CI operators.Th 3.2 nontrivial projections are CI operators.Th 3.3 If (A), and A has SVEP at A, then A - A is a CI operator.Th 3.4 If T is an invertible operator , A is a CI operator , then both TA and AT are CI operators.Th 3.6 If T is an invertible operator , A is not a CI operator , then both TA and AT are not CI operators.Th 3.8 If T KA(H), then both T and T are CI operators. Th 3.9 If T KA(H), then (T), for every C , 0.This part also give the characterization of CI operators in terms of Mbekhta's subspaces which are similar to that in paper [3].A is a CI operator if and only if one of the following three statements exists.(a) A is invertible.(b) H0(A) = 0 and K(A) H.(c) H0(A) = 0 , K(A) H and R(A) is not closed .In the last part we discuss the relationship between the generalized inverse of an operator and Mbekhta's subspaces and thet between CI operators and the generalized inverse of an operator .The main results are expressed bellow.1 If A inv B then AB is CI opertor if and if only if one of the three statements exists .(a) R(A) is not closed .(b) R(A) = R(A) = H and N(B)=0.(c) R(A) = R(A) C H and N(B) = 0.2 If A inv B ,R(A) = R(A) H and N(A) = 0 , R(B) = R(B) H , N(B) = 0. then AB and BA are CI operators.3 If A inv B then K(A) N(a-A) K(AB).4 If A inv B then K(A) = K(AB) = N(X - A).5 If A inv B then H0(AB)=N(AB).