A note on the Drazin inverses with Banachiewicz–Schur forms

Abstract

Let H be a Hilbert space, M the closed subspace of H with orthocomplement M ⊥ . According to the orthogonal decomposition H = M ⊕ M ⊥ , every operator M ∈ B ( H ) can be written in a block-form M = A B C D . In this note, we give necessary and sufficient conditions for a partitioned operator matrix M to have the Drazin inverse with Banachiewicz–Schur form. In addition, this paper investigates the relations among the Drazin inverse, the Moore–Penrose inverse and the group inverse when they can be expressed in the Banachiewicz–Schur forms.