Further results on the $(b, c)$-inverse, the outer inverse $A^{(2)}_{T, S}$ and the Moore-Penrose inverse in the Banach context

TL;DR

This paper explores advanced inverse concepts like the $(b, c)$-inverse, outer inverse, and Moore-Penrose inverse within Banach space operators, focusing on their properties such as continuity, differentiability, and invertibility sets.

Contribution

It provides new insights into the properties and relationships of these generalized inverses in Banach algebra contexts, extending previous results.

Findings

Analysis of continuity and differentiability of inverse sets

Characterization of relationships between different inverse types

Identification of conditions for invertibility and openness

Abstract

In this article properties of the $(b, c)$-inverse, the inverse along an element, the outer inverse with prescribed range and null space $A^{(2)}_{T, S}$ and the Moore-Penrose inverse will be studied in the contexts of Banach spaces operators, Banach algebras and $C^*$-algebras. The main properties to be considered are the continuity, the differentiability and the openness of the sets of all invertible elements defined by all the aforementioned outer inverses but the Moore-Penrose inverse. The relationship between the $(b, c)$-inverse and the outer inverse $A^{(2)}_{T, S}$ will be also characterized.