Smoothing of operator semigroups under relatively bounded perturbations

TL;DR

This work develops a general perturbation framework for smoothing properties of operator semigroups, showing that ultracontractivity relative to a Banach space $V$ is preserved under Miyadera–Voigt-type perturbations. It then proves analytic perturbation results for the spectrum, establishing analytic dependence of eigenvalues $\lambda(\kappa)$ and spectral projections $P(\kappa)$ in $V$, with preserved lower bounds on eigenfunctions in Banach lattices. The theory is then applied to elliptic operators on domains, including second- and higher-order operators, fractional powers, and nonlocal perturbations, yielding concrete smoothing, spectral, and eventual-positivity results. Overall, the paper provides a robust abstract toolkit for long-time dynamics of PDEs under broad perturbations and connects semigroup smoothing to spectral and positivity properties in concrete elliptic problems.

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.

Theorems & Definitions (40)

• Theorem 1.1: Ultracontractivity under bounded perturbations
• Theorem 1.2
• Proposition 2.3
• Remark 2.4
• Theorem 2.5
• Lemma 2.6
• proof : Proof of Lemma \ref{['lem:V-bounded']}
• Lemma 2.7
• proof
• proof : Proof of Theorem \ref{['thm:perturb-smoothing']}