A note on formulae for the generalized Drazin inverse of anti-triangular block operator matrices in Banach spaces

In this paper, we separately obtain new explicit expressions for the generalized Drazin inverse of anti-triangular block operator matrices under certain restrictions. As applications, we utilize the relationship between the anti-triangular block operator matrix and a 2×2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\times 2$$\end{document} block operator matrix to establish several formulae for the generalized Drazin inverse of a 2×2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\times 2$$\end{document} block operator matrix. Our results generalize and unify a series of results about the Drazin inverse and the generalized Drazin inverse in the literature. We give certain numerical examples to illustrate our results.