generalized inverse of block operator matrices over banach algebras

We introduce a new kind of generalized inverse which is called $pi-$hirano inverse‎. ‎an element $ain mathcal{a}$ has $pi-$hirano inverse if there exists $xin mathcal{a}$ such that $x in comm(a)$‎, ‎$x=xax$ and $a-a^{n+2}xin n(mathcal{a})$ for some $nin {bbb n}$‎. ‎in this paper‎, ‎some elementary properties of the $pi-$hirano inverse are obtained‎. ‎we investigate the existence of the $pi-$hirano inverse for the anti-triangular operator matrix $n=left[ egin{array}{cc} a&bc&0 end{array} ight]$ with $abc=0$ or $bca=0$ and $ca^2=0$‎. ‎certain multiplicative and additive results for the $pi-$hirano inverse in a banach algebra are presented‎.