Research On The Quasi-inverse And Generalized Inverse Of Operators

Operator algebra and operator theory which is an important re-search field in functional analysis involves many branches of basic mathematics and applied mathematics,such as algebra,geometry,matrix theory,optimization theory and quantum physics.Operator spectrum theory is one of the active re-search topics in operator theory.The invertibility of operators is concerned by more and more scholars.Many scholars have studied the classification of opera-tor algebra by considering the invertibility of operators as isomorphic invariants and obtained lots of results.Some scholars have studied the preserver problem of quasi-invertibility(that is,invertibility under quasi-product)of operators.The generalized inverse is another concept of operators.The generalized inverse of an operator has a wide variety and different applications.Therefore,there is a lot of space to study the generalized inverse of operators.In this paper,we study additive maps preserving quasi-invertibility or quasi-zero factors on operator algebra and the properties and applications of generalized inverse of operators.There are three aspects to the specific study.In chapter 2,let B(X),B(Y)be the set of all bounded linear operators on the complex Banach space X,Y with dimentions greater than 1,respectively.Let A,B be normed closed subalgebras of B(X),B(Y)containing finite rank operators,re-spectively.A characterization of additive surjective mappings from A onto B which preserve any one of(left,right)quasi-invertibility and(left,right,semi)quasi-zero divisors in both directions is given.In chapter 3,let B(H)be the set of all bounded linear operators on the com-plex Hilbert space H.By using space decomposition and block operator matrices in different decomposition of Hilbert space,we give some necessary and suffcient con-ditions for the existence of Re-nnd {1}-,{1,3}-,{1,4}-inverses,hermitian {1,3}-,{1,4}-inverses and nonnegative {1,3}-,{1,2,3}-,{1,4}-,{1,2,4}-,{1,3,4}-inverses for an operator on B(H)with closed range.In chapter 4,using the generalized inverse of operators,we give some equiva-lent descriptions of generalized projection and hypergeneralized projection,and the product of two oblique projections to be also oblique projection.Then we study generalization Kovarik formula of oblique projections pairs.Besides,we character-ize*-partial order by using generalized inverse of operators.