GENERALIZED INVERSES OF ANTI-TRIANGULAR BLOCK OPERATOR MATRICES

We introduce a new class of generalized inverse which is called π-Hirano inverse‎. Let A be a Banach algebra with an identity. We first recall the definitions of some generalized inverses. As is well known, in 1958, Drazin [6] defined, an element a∈A has Drazin inverse if there is the element a∈A which satisfies ax=xa, xax=x and a-a^2 x∈N(A). In this paper some elementary properties of the π-Hirano inverse are obtained‎. ‎We investigate the existence of the π-Hirano inverse for the anti-triangular operator matrix N=[0&B&C&D] with DCB=0 and et al‎. ‎Certain multiplicative and additive results for the π-Hirano inverse in a Banach algebra are presented‎. ‎We then apply some conditions under which a 2×2 block operator matrix has π-Hirano inverse over Banach spaces‎.