The g-Drazin inverses of anti-triangular block operator matrices

TL;DR

This paper derives explicit formulas for the g-Drazin inverse of certain anti-triangular block operator matrices, enabling solutions to broader classes of singular differential equations.

Contribution

It introduces new explicit representations of the g-Drazin inverse for specific block operator matrices, extending previous methods.

Findings

Explicit formulas for g-Drazin inverses of block matrices

Application to solving singular differential equations

Extension of existing inverse computation techniques

Abstract

An element $a$ in a Banach algebra $\mathcal{A}$ has g-Drazin inverse if there exists $b\in \mathcal{A}$ such that $ab=ba, b=bab$ and $a-a^2b \in \mathcal{A}^{qnil}$. In this paper we find new explicit representations of the g-Drazin inverse of the block operator matrix $\left( \begin{array}{cc} E&I F&0 \end{array} \right)$. We thereby solve a wider kind of singular differential equations posed by Campbell [S.L. Campbell, The Drazin inverse and systems of second order linear differential equations, Linear $\&$ Multilinear Algebra, 14(1983), 195--198].