Acute and Stable Perturbations of the Drazin Inverse of Bounded Linear Operators in Banach Spaces. (25th October 2020) - Ma, Haifeng
Abstract: Let B   ( X ) be the set of bounded linear operators on a Banach space X, and a liner operator A ∈ B   ( X ) be Drazin invertible. A linear operator B ∈ B   ( X ) is said to be a stable perturbation of A if B is Drazin invertible and I − A π − B π is invertible, where I is the identity operator on X, A π ( = I − A A D ) and B π ( = I − B B D ) 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 ρ ( B π − A π ) < 1 . Under the similar condition that a matrix B is a stable perturbation of a matrix A, an explicit formula for the Drazin inverse B D 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 B π − A π is generalized from the finite-dimensional case to the Banach space.