A Class Of Block Operator Matrices Drazin Inverse Said Its Indicators

The theory of the Drazin inverse has been a substantial growth over the past decades. It is applicated in many diverse areas including statistics, numerical analysis, differential equations, Markov chains, population models, cryptography, and control theory, etc. (see [2,9,12]).One important research field of the Drazin inverse is its representation theory. In recent years, using the decomposition of matrices, many people have derived the representation for the Drazin inverse of a block operator matrix. Yet due to the lack of valid methods, little has been done on the explicit characterization of the Drazin index.Base on the theory of stable perturbation, we not only give a new approach to providing the representation for the Drazin inverse of a block operator matrix, but also to obtaining the explicit characterization of the Drazin index. In order to compare the difference between the technique of matrix decomposition and our approach, we do that. Firstly, using the decomposition of operator matrices, we obtain the explicit representations for the Drazin inverse of a 2×2 block matrix under certain conditions. However, we can not provide the concrete Drazin index. Then, base on the perturbation theory of the Drazin inverse, we get the explicit representations for the Drazin inverse of certain 2×2 block operator matrix. Furthermore, we obtain the associated concrete Drazin index.This paper consists of three parts. In Chapter 1, we recall some notations and definitions of the Drazin inverse. We describe the motivation, the difficulties and the main results of this paper. In Chapter 2, inspired by the recent work of Dragana S. Cvetkovic-Ilic [22], in this paper we give a new representation for the Drazin inverse of a 2×2 block matrix under certain conditions by the technique of matrix decomposition. However, the limitation of this technique is that one can not obtain the concrete Drazin index.In Chapter 3, we introduce a new technique to get the representation for the Drazin inverse of a block operator matrix. Firstly, given two bounded linear operators F, G on a Banach space X such that G2F=GF2=0, we get the representation of the Drazin inverse of the operator matrix in the form Inspired by the paper [31], we find M is a stable perturbation of M, where Then an explicit formula for the Drazin inverse MD is derived. In particular, we provide the Drazin index of M under certain conditions. Finally, using the obtained results, we provide the representation of the Drazin inverse of F+G.