Acute and Stable Perturbations of the Drazin Inverse of Bounded Linear Operators in Banach Spaces. (25th October 2020)

• Record Type: Journal Article
• Authors: Ma, Haifeng
• Is Part Of: Numerical functional analysis and optimization. Volume 41:Number 14(2020)
• Page Start: 1748
• Page End: 1760
• Publication Date: 2020-10-25
• Subjects: Acute perturbation -- Banach space -- Drazin index -- Drazin inverse -- group inverse -- stable perturbation; 15A09 -- 47A05 -- 47A55; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2020.1813162
• Languages: English
• ISSNs: 0163-0563

Abstract

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.