Mbekhta's Subspaces And Invertibility Of Operators

Two important subspaces introduced by M.Mbekhta in 1987[4].(They are usually called Mbekhta subspaces .)In the following years,Mbekhta subspaces have been widely used in spectral theory of bounded operators and compact operators,the single-valued extension property (SVEP) of bounded operators ,and so on. Professor Gongwei Bang introduced the concept of CI(consistent in invertibility) operator and gave the necessary and sufficent conditions of CI operator in 1994.The results that are quoted in this paper are classical conclusions.On the basis of the results .this paper discusses the invertibility of operators.CI operators and generalizes .some results about spectral theory of bounded operators and properties of Mbekhta subspaces in terms of Mbekhta subspaces .The first section quotes some concepts and results that will be used in this paper.On the basis of [7.corollary 1.3].I aquire a theorem as follows:Theorem 1.18 Suppose ..Then I) is the only isolated point in a(A). Theorem 1.19 Suppose Then A0 is the only isolated point, in (A).Corollary 1.20 Suppose Then there is no isolated point in (A). Corollary 1.21 Supppose Then there is no isolated point in (A).The second section discusses properties of left-invertible and right-invertible operator in use of Mbekhta's subspaces and gives the necessary and sufficent condition.I get results as follows:Theorem 2.6 Suppose A B(X) is the left-invertible . Then 1). ,Anis left-invertible . 2).H0(A) = {0}.3).K(A) = R(An) is closed . and if X is Hilbert space.then: 4). A* is right-invertible . 5). A is regular .Theorem 2.7 Suppose ,A B(X) is right-invertible. and it X is Hilbert space.then 4).A* is left-invertible, .5). A is regular . Corollary 2.8 Suppose A B(X).Then A is invertible if and only if H0(A) = {0}, K(A) = X.The third section gives some results about judgement CI operator.The first part discusses judgement that A is a CI operator on the basis of [5, Theorem 1.1] and gives results as follows:Theorem 3.5 A B(H). If K(A) is not closed, then A is a CI operator. Theorem 3.10 A 6 B(H).If K(A) = K(A*),then (AA*) = (A*A).so A is a CI operator.The second part gives the necessary and sufficent condition on the basis of [1, VII, Prop. 6.4] and gets theorem as follows:Theorem 3.7 A B(H). .then is a CI operator if and only if 2)If (A). then .4 - A is a CI operator if and only if R(A - A) is not.closed or The last section gives some examples of operators that satisfies following equal-ity formulas:gets results as follows:Example 1.4: Invertible operators satisfies 1.1) and 1.2) at then same time. Kxample 4.8: P B(X) is a nontrival projection.then it. satisfies 4.1) and 4.2).