The generalized Drazin inverse of operator matrices

Authors : Li GUO, Honglin ZOU, Jianlong CHEN

Pages : 1134-1149

Doi:10.15672/hujms.731518

View : 16 | Download : 6

Publication Date : 2020-06-02

Article Type : Research Paper

Abstract :Representations for the generalized Drazin inverse of an operator matrix $\begin{pmatrix}A & B \\ C & D \end{pmatrix}$ are presented in terms of $A,B,C,D$ and the generalized Drazin inverses of $A,D$, under the condition that $BD^d=0,~\text{and}~BD^iC=0,~\text{for any nonnegative integer}~ i.$ Using the representation, we give a new additive result of the generalized Drazin inverse for two bounded linear operators $P,Q \in \mathbf{B}insert ignore into journalissuearticles values(X);$ with $PQ^{d}=0$ and $PQ^{i}P=0$, for any integer $i\geq 1$. As corollaries, several well-known results are generalized.
Keywords : Banach space, generalized Drazin inverse, operator matrix