Some Generalized Inverses Of Rings And Algebras With Involution

Generalized inverse has a widespread prospect on matrix,operator theory,d-ifferential equation,numerical analysis and Markov chain,and it also has acquired various significant applications,such as statistics,cryptography,cybernetics and coding theory.In this thesis,we study some generalized inverses of rings and alge-bras with involution.The thesis is organized as follows:In Chapter 1,we state the research background and our main work of the current thesis.In Chapter 2,we recall some preliminaries.In Chapter 3,we firstly discuss the relation among the strongly left(b,c)-invertibility of a,the right ca-regularity of b,and the(b,c)-invertibility of a in rings,and we also describe the right(resp.left)c-regular elements in quasi-normal rings or in directly finite rings.Secondly,we give a novel characterization of group inverses(resp.MP-inverses,resp.EP elements)in rings with involution by means of right(resp.left)c-regular elements.Finally,we discuss a strongly left(b,c)-invertible element of a ring to be(b,c)-invertible,and prove that if a ring satisfies that for any left non-annihilator of left minimal idempotent e is(e,e)-invertible,then it is a left min-Abel ring.In Chapter 4,we describe the elements in rings with involution which are EP.Especially,we reduce the preliminary requirements met in some existing results.In addition,contact is established between EP elements(resp.partial isometry,resp.normal EP elements,resp.strongly EP elements)and the solutions of certain equations.In Chapter 5,based on the results of weighted MP-inverse in C*-algebra,we explore the existence of MP-inverse in rings with involution.In particular,an element a of a ring with involution is MP-invertible if and only if it is specially(a*,a*)-regular.In addition,we give some equivalent conditions for an element of a C*-algebra to be a weighted-EP element and weighted partial isometry.For example,a regular element a of a C*-algebra is a weighted partial isometry w.r.t(e,f)if and only if aa*f,e is an idempotent,if and only if a*f,e a is an idempotet,where e,f are positive invertible elements.Finally,we extend the results of weighted partial isometry in C*-algebras to even-order tensors over complex numbers.In Chapter 6,motivated by the reverse order laws for the MP-inverse in rings with involution and the weighted MP-inverse in C*-algebras,we study several novel equivalent conditions for reverse order laws for the MP-inverse and the weighted MP-inverse in even-order tensors over complex numbers.Specifically,for any two even-order tensors A and B of order 2K,the reverse order law for the MP-inverse of A and B holds if and only if B(?)(?)KA(?)is a {1,3,4}-inverse of tensor A*K B,and the reverse order law for the weighted MP-inverse of A and B is satisfied if and only if BNM(?)*K APN(?)(?)is a weighted {1,3,4}-inv*erse of tensor A*K B with weights P and M,where K is a nonnegative integer and M,P,N are Hermitian positive definite.