Microspectral analysis of quasinilpotent operators

TL;DR

This paper introduces a microspectral theory for quasinilpotent operators in Banach spaces, revealing deeper structural insights beyond classical spectral theory, especially for non-compact, non-normal, and non-nilpotent cases.

Contribution

It develops a new microspectral framework for quasinilpotent operators, providing detailed analysis of their properties through microspectral sets in the complex plane.

Findings

Microspectral sets characterize semigroup generation.

Microspectral analysis informs resolvent behavior.

Results apply to non-compact, non-normal, non-nilpotent operators.

Abstract

We develop a microspectral theory for quasinilpotent linear operators $Q$ (i.e., those with $\sigma(Q) = \{0}$) in a Banach space. When such $Q$ is not compact, normal, or nilpotent, the classical spectral theory gives little information, and a somewhat deeper structure can be recovered from microspectral sets in $\C$. Such sets describe, e.g., semigroup generation, resolvent properties, power boundedness as well as Tauberian properties associated to $zQ$ for $z \in \C$.