Smoothing of operator semigroups under relatively bounded perturbations

TL;DR

This paper studies how certain smoothing properties of operator semigroups are preserved under specific perturbations, leading to new spectral and positivity results for evolution equations.

Contribution

It establishes the stability of smoothing properties under relatively bounded perturbations and derives new spectral and positivity theorems for operator semigroups.

Findings

Smoothing properties are stable under certain perturbations.

Spectral perturbation theorems are established.

Results have implications for long-term dynamics of elliptic evolution equations.

Abstract

We investigate a smoothing property for strongly-continuous operator semigroups, akin to ultracontractivity in parabolic evolution equations. Specifically, we establish the stability of this property under certain relatively bounded perturbations of the semigroup generator. This result yields a spectral perturbation theorem, which has implications for the long-term dynamics of evolution equations driven by elliptic operators of second and higher orders. In particular, a new perturbation theorem for so-called eventually positive semigroups is derived as a consequence of the general results.