New Representations of the g-Drazin Inverse for Anti-Triangular Matrices

Abstract

We provide representations for the generalized Drazin inverse of an anti-triangular matrix of the form
$\left(
\begin{array}{cc}
a&b\\
1&0
\end{array}
\right)$ in a Banach algebra $\mathcal{A}$ with commuting $a, b\in \mathcal{A}$. In particular, a new formula for the Drazin inverse of an anti-triangular matrix is established by employing Catalan numbers.