Microspectral analysis of quasinilpotent operators

We develop a microspectral theory for quasinilpotent linear operators Q (i.e., those with σ(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 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∈C.