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. تفاصيل المقالة

Matrix analysis is one of the appropriate mathematical methods used in engineering sciences. In methods based on matrix analysis special inverse matrix , properties of matrices are used in linear algebra problems. For the first time, the structure of matrix converters was presented by gyagy and pully under the purely theoretical concept with the motivation of optimizing the fundamental performance of cyclo-converters, in order to obtain output voltage with unlimited frequency and direct conversion using controllable two-way switches. Considering the essential role of inverse matrices in engineering sciences and considering the class of matrices inverses. Now, We introduce a new class of generalized inverse which is called n−strongly Drazin inverse and some elementary properties of the n−strongly Drazin inverse are obtained. Certain multiplicative and additive results for the n−strongly Drazin inverse in a Banach algebra are presented. We then apply some conditions under which a 2×2 block operator matrix has n−strongly Drazin inverse over Banach spaces. تفاصيل المقالة