Drazin invertibility of linear operators on quaternionic Banach spaces

TL;DR

This paper investigates the Drazin inverse for right linear operators on quaternionic Banach spaces, providing explicit formulas and extending known complex operator results to the quaternionic setting.

Contribution

It introduces a characterization of the Drazin inverse in quaternionic Banach spaces and extends classical results from complex Banach space theory.

Findings

Drazin inverse exists under certain spectral conditions

Explicit formula for the Drazin inverse involving a function f

Extension of complex operator results to quaternionic context

Abstract

Let $A$ be a right linear operator on a two-sided quaternionic Banach space $X$. The paper studies the Drazin inverse for right linear operators on a quaternionic Banach space. It is shown that if $A$ is Drazin invertible then the Drazin inverse of $A$ is given by $f(A)$ where $f$ is $0$ in an axially symmetric neighborhood of $0$ and $f(q) = q^{-1}$ in an axially symmetric neighborhood of the nonzero spherical spectrum of $A$. Some results analogous to the ones concerning the Drazin inverse of operators on complex Banach spaces are proved in the quaternionic context.