A New Kind Of Generalized Inverse And Its Applications

Generalized inverses and their applications have developed rapidly in the 20 th century,and have become an important research content in matrix theory and operator theory.In this paper,Mosic-Abyzov inverse on Banach algebras is introduced and studied,and then the decomposition of matrices and generalized inverses are linked.We extend the Cline formula of the generalized Drazin inverse to the Mosic-Abyzov inverse of the matrix,thus obtaining the existence conditions of the Mosic-Abyzov inverse of the element and matrix,and characterizing the new decomposition properties of the operator matrix.The specific structure of this article is as follows:In Chapter 1,we introduce the research background,basic definition and some common conclusions involved in this paper.In Chapter 2,in a Banach algebra,we study the relationship between Mosic-Abyzov inverse and generalized Drazin inverse,the equivalent conditions for the existence of Mosic-Abyzov inverse,and obtain the Cline formula of Mosic-Abyzov inverse,which provides a theoretical basis for the following research.In Chapter 3,we discuss the additive properties of Mosic-Abyzov inverse of elements in Banach algebras under orthogonal conditions and commutative conditions.In Chapter 4,the Mosic-Abyzov inverse of anti-triangular torsion matrix in Banach algebras is studied by using the solvability of a class of equations.In Chapter 5,the Mosic-Abyzov inverse of the sum and product of operator matrices in Banach spaces is investigated by using the Peirce decomposition.In Chapter 6,we summarize the research results of Mosic-Abyzov inverse and look forward to its development prospects.