Representations for generalized Drazin inverse of operator matrices over a Banach space

In this paper we give expressions for the generalized Drazin inverse of a (2,2,0) operator matrix and a $2\times2$ operator matrix under certain circumstances, which generalizes and unifies several results in the literature

ADDITIVE PROPERTIES OF THE DRAZIN INVERSE FOR MATRICES AND BLOCK REPRESENTATIONS: A SURVEY

In this paper, a review of a development of the Drazin inverse for the sum of two matrices has been given. Since this topic is closely related to the problem of finding the Drazin inverse of a 2x2 block matrix, the paper also offers a survey of this subject

Expressions for the g-Drazin inverse of additive perturbed elements in a Banach algebra

AbstractWe study additive properties of the g-Drazin inverse in a Banach algebra A. In our development we derive a representation of the resolvent of a 2×2 matrix with entries in A, which is then used to find explicit expressions for the g-Drazin inverse of the sum a+b, under new conditions on a,b∈A. As an application of our results we obtain a representation for the Drazin inverse of a 2×2 complex block matrix in terms of the individual blocks, under certain conditions

On Drazin inverse of operator matrices

AbstractIn this short paper, we offer (another) formula for the Drazin inverse of an operator matrix for which certain products of the entries vanish. We also give formula for the Drazin inverse of the sum of two operators under special conditions

Representations for the Drazin inverse of the sum P+Q+R+S and its applications

AbstractLet P, Q, R and S be complex square matrices and M=P+Q+R+S. A quadruple (P,Q,R,S) is called a pseudo-block decomposition of M ifPQ=QP=0PS=SQ=QR=RP=0andRD=SD=0,where RD and SD are the Drazin inverses of R and S, respectively. We investigate the problem of finding formulae for the Drazin inverse of M. The explicit representations for the Drazin inverses of M and P+Q+R are developed, under some assumptions. As its application, some representations are presented for 2×2 block matricesAB0CandABDC, where the blocks A and C are square matrices. Several results of this paper extend the well known representation for the Drazin inverse ofAB0Cgiven by Hartwig and Shoaf, Meyer and Rose in 1977. An illustrative example is given to verify our new representations

Rank Equalities Related to Generalized Inverses of Matrices and Their Applications

This paper is divided into two parts. In the first part, we develop a general method for expressing ranks of matrix expressions that involve Moore-Penrose inverses, group inverses, Drazin inverses, as well as weighted Moore-Penrose inverses of matrices. Through this method we establish a variety of valuable rank equalities related to generalized inverses of matrices mentioned above. Using them, we characterize many matrix equalities in the theory of generalized inverses of matrices and their applications. In the second part, we consider maximal and minimal possible ranks of matrix expressions that involve variant matrices, the fundamental work is concerning extreme ranks of the two linear matrix expressions $A - BXC$ and $A - B_1X_1C_1 - B_2X_2C_2$. As applications, we present a wide range of their consequences and applications in matrix theory.Comment: 245 pages, LaTe