Microspectral analysis of quasinilpotent operators
Volume 112 / 2017
Banach Center Publications 112 (2017), 281-306
MSC: 47A10, 47B06, 47B10.
DOI: 10.4064/bc112-0-15
Abstract
We develop a microspectral theory for quasinilpotent linear operators $Q$ (i.e., those with $\sigma(Q) = \{0 \}$) in a Banach space. For such operators, the classical spectral theory gives little information. Deeper structure can be obtained from microspectral sets in $\mathbb C$ as defined below. Such sets describe, e.g., semigroup generation, various resolvent properties, power boundedness as well as Tauberian properties associated to $zQ$ for $z \in \mathbb C$.
