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

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 $a-a^{2}b\in {\mathcal{A}}^{qnil}$. In this paper we find new explicit representations of the g-Drazin inverse of the block operator matrix $\left(\begin{array}{ccc}E& IF& 0\end{array}\right)$. 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 $\mathrm{\&}$ Multilinear Algebra, 14(1983), 195–198].