Abstract
In a Banach algebra A, let x=[abcd]∈A relative to the idempotent p∈A, where a∈pAp is generalized Drazin invertible. Under assumptions that the generalized Schur complement s=d−cadb∈(1−p)A(1−p) and the element caπb∈(1−p)A(1−p) are generalized Drazin invertible, we establish some formulae for the generalized Drazin inverse of x in terms of a matrix in the generalized Banachiewicz–Schur form and its powers. We develop necessary and sufficient conditions for the existence and the expressions for the group inverse of a block matrix in Banach algebras. The provided results extend earlier works given in the literature.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
