Sharp growth rates for semigroups using resolvent bounds

TL;DR

This paper establishes conditions under which resolvent bounds imply specific growth rates for semigroups, with applications to spectral analysis and optimality in wave equations.

Contribution

It provides new links between resolvent bounds and semigroup growth rates, extending results to various Banach spaces and fractional domains.

Findings

Resolvent bounds on imaginary lines imply semigroup growth rates under certain conditions.

Main results are applicable to Hilbert spaces, asymptotically analytic, positive semigroups, and more general Banach spaces.

Application to a classical wave equation example demonstrates the optimality of the results.

Abstract

We study growth rates for strongly continuous semigroups. We prove that a growth rate for the resolvent on imaginary lines implies a corresponding growth rate for the semigroup if either the underlying space is a Hilbert space, or the semigroup is asymptotically analytic, or if the semigroup is positive and the underlying space is an $L^{p}$-space or a space of continuous functions. We also prove variations of the main results on fractional domains; these are valid on more general Banach spaces. In the second part of the article we apply our main theorem to prove optimality in a classical example by Renardy of a perturbed wave equation which exhibits unusual spectral behavior.