The Research On The Mary Inverse And The Generalized Drazin Inverse

The Mary inverse and the generalized Drazin inverse are two important types of generalized inverses,which have extensive applications in many fields.In this thesis,the Mary inverse in semigroups and rings,the Drazin inverse of matrices over rings,the pseudo Drazin inverse and the generalized Drazin inverse in Banach algebras are studied.The main contents are arranged as follows:In Chapter 2,we mainly study the Mary inverse in a ring.First of all,some existence criteria for the Mary inverse are obtained by means of idempotents,invertibility of certain elements,one-sided annihilators ideals and one-sided principal ideals in a ring.Then,we consider the necessary and sufficient conditions and representations for the inner Mary inverse,which unify the related results on the group inverse and Moore-Penrose inverse.In Chapter 3,we mainly investigate the Mary inverse in a semigroup.First,the charac-terizations and representations of the Mary inverse are given.Then,we consider the Cline’s formula for the Mary inverse and the Mary inverse of the product,which unify the related results about the classic generalized inverses.At last,the relation between the core inverse,dual core inverse and the Mary inverse are studied,which improve the results obtained by D.S.Rakic et al.for the core inverse,dual core inverse.In Chapter 4,we are concerned with the existence and representation for the Drazin inverse of matrices over a ring.First,we study the expression for the anti-triangular matrix M=[abc0]under some conditions,which extend the relevant results for the complexmatrices and operator matrices investigated by C.Y.Deng,C.J.Bu,J.J.Huang et al.Then,we take advantage of the generalized Schur complement to discuss the existence,expression and index for the Drazin inverse of block matrices over a ring,which improve the works given by R.E.Hartwig et al.for the Draszin inverse.In Chapter 5,we consider the pseudo Drazin inverse in a Banach algebra.First,the matrix representation of a pseudo Drazin invertible element,the expression for the pseudo Drazin inverse of the triangular matrix,as well as the formula for the sum of two pseudo Drazin invertible elements are obtained,which generalize the related results given by Z.Wang and J.L.Chen.Then,we use the centralizer to study the existence and representation for the product and sum of two pseudo Drazin invertible elements,which extend the relevant works investigated by H.H.Zhu and J.L.Chen.In chapter 6,the equivalent conditions for the existence and the expression of the general-ized Drazin inverse of a + b are obtained,where both a and b are generalized Drazin invertible elements satisfying a2b = aba and b2a = bab;meanwhile,the representation for the general-ized Drazin inverse of ab is investigated,which improve the results given by J.J.Koliha,D.S.Cvetkovic-Ilic et al.for the generalized Drazin inverse.As an application of additive results;we obtain some new representations for the generalized Drazin inverse of a block matrix in a Banach algebra.