Hirano inverse of anti-triangular matrix over Banach Algebras

Abstract

In this paper we investigate Hirano invertibility of anti-triangular matrix over a Banach algebra. Let $a\in {\mathcal{A}}^H,b\in {\mathcal{A}}^{sD}.$ If $b^{D}a=0,ba{b}^{\pi }=0,$ we prove that $\left(\begin{array}{cc}a& 1\\ b& 0\end{array}\right)\in M_2(\mathcal{A}{)}^H.$ Moreover, we considered Hirano invertibility of anti-triangular matrices under commutative-like conditions. These provide new kind of operator matrices with tripotent and nilpotent decompositions.

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