The $g$-Drazin Invertibility of a Block Operator Matrix

We present new additive properties of the $g$-Drazin inverse of a linear operator on a Banach space. The $g$-Drazin invertibility of certain $2\times 2$ block operator matrices on a Banach space is thereby established. These results extend many known results, e.g., by Yang and Liu [J. Comput. Applied Math. 235, 1412–1417 (2011)] and Dopazo and Martinez-Serrano [Linear Algebra Appl. 432, 1896–1904 (2010)].