Study on Hirano Invertibility of Two New Types of Perturbed Operator Matrices

We investigate the Hirano invertibility of block-operator matrices in Banach algebras, and obtain the Hirano inverse of matrix $\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]$ under two types of new perturbation conditions. Furthermore, we provide a new operator matrix decomposed into the sum of tripotent and nilpotent elements on Banach spaces.