Microspectral analysis of quasinilpotent operators

Abstract

We develop a microspectral theory for quasinilpotent linear operators $Q$ (i.e., those with $σ(Q) = \{0}$) 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 $z \in \C$.