Generalized inverses and the maximal radius of regularity of a Fredholm operator

TL;DR

This paper investigates operators with analytic generalized inverses, characterizes their properties, and computes the maximal radius of regularity for Fredholm operators using spectral radius, addressing a conjecture by Zemánek.

Contribution

It introduces new characterizations of operators with analytic generalized inverses and computes the maximal radius of regularity for Fredholm operators in terms of spectral radius.

Findings

Operators with analytic generalized inverses satisfy the resolvent identity.

The maximal radius of regularity for Fredholm operators is expressed via spectral radius.

Partial resolution of Zemánek's conjecture on the radius of regularity.

Abstract

Operators possessing analytic generalized inverses satisfying the resolvent identity are studied. Several characterizations and necessary conditions are obtained. The maximal radius of regularity for a Fredholm operator T is computed in terms of the spectral radius of a generalized inverse of T. This provides a partial answer to a conjecture of J. Zem\'anek.