Characterizations for the n-strong Drazin invertibility in a ring

We introduce a new class of generalized inverse which is called ??Hirano inverse. In this paper some elementary properties of the ??Hirano inverse are obtained. We prove that a ? A is ??Hirano invertible if and only if a ? an+1 is nilpotent for some positive integer n. Certain multiplicative and additive results for the ??Hirano inverse in a Banach algebra are presented. We then apply these new results to block operator matrices over Banach spaces.

An extension of Hirano inverses in Banach algebras

Ghaffari

Haddadi

Abdolyousefi

2022

Extensions of G-outer inverses

Mosic,

Stanimirovic,

Ciric

2023

Our first objective is to present equivalent conditions for the solvability of the system of matrix equations ADA = A, D= B and CAD = C, where D is unknown, A, B,C are of appropriate dimensions, and to obtain its general solution in terms of appropriate inner inverses. Our leading idea is to find characterizations and representations of a subclass of inner inverses that satisfy some properties of outer inverses. A G-(B,C) inverse of A is defined as a solution of this matrix system. In this way, G-(B,C) inverses are defined and investigated as an extension of G-outer inverses. One-sided versions of G-(B,C) inverse are introduced as weaker kinds of G-(B,C) inverses and generalizations of one-sided versions of G-outer inverse. Applying the G-(B,C) inverse and its one-sided versions, we propose three new partial orders on the set of complex matrices. These new partial orders extend the concepts of G-outer (T, S)-partial order and one-sided G-outer (T, S)-partial orders.

Extension of the generalized n-strong Drazin inverse

Mosic,

Zou,

Wang

2023

The aim of this paper is to present an extension of the generalized n-strong Drazin inverse for Banach algebra elements using a g-Drazin invertible element rather than a quasinilpotent element in the definition of the generalized n-strong Drazin inverse. Thus, we introduce a new class of generalized inverses which is a wider class than the classes of the generalized n-strong Drazin inverse and the extended generalized strong Drazin inverses. We prove a number of characterizations for this new inverse and some of them are based on idempotents and tripotents. Several generalizations of Cline?s formula are investigated for the extension of the generalized n-strong Drazin inverse.