Acute and Stable Perturbations of the Drazin Inverse of Bounded Linear Operators in Banach Spaces

Abstract

Let be the set of bounded linear operators on a Banach space X, and a liner operator be Drazin invertible. A linear operator is said to be a stable perturbation of A if B is Drazin invertible and is invertible, where I is the identity operator on X, and are the spectral projectors of A and B respectively. We call B an acute perturbation of A with respect to the Drazin inverse if the spectral radius Under the similar condition that a matrix B is a stable perturbation of a matrix A, an explicit formula for the Drazin inverse BD is derived by Xu, Song and Wei (The stable perturbation of the Drazin inverse of the square matrices, SIAM J. Matrix Anal. Appl., 31(3) (2010), pp. 1507-1520). This formula is generalized to the infinite-dimensional case, and some new formulas for spectral radius of is generalized from the finite-dimensional case to the Banach space.