Decay of Operator Semigroups, Infinite-time Admissibility, and Related Resolvent Estimates

TL;DR

This paper investigates how decay rates of bounded $C_0$-semigroups relate to resolvent estimates and $L^p$-admissibility, providing new characterizations and conditions in Hilbert and $L^q$-space settings.

Contribution

It offers novel characterizations of polynomial decay via resolvent behavior and $L^p$-admissibility, including sufficient conditions for $L^2$-admissibility in polynomially stable semigroups.

Findings

Polynomial decay characterized by resolvent behavior in right half-plane.

$L^p$-admissibility characterizes decay for multiplication semigroups on $L^q$-spaces.

Sufficient condition for $L^2$-admissibility in polynomially stable Hilbert space semigroups.

Abstract

We study decay rates for bounded $C_0$-semigroups from the perspective of $L^p$-infinite-time admissibility and related resolvent estimates. In the Hilbert space setting, polynomial decay of semigroup orbits is characterized by the resolvent behavior in the open right half-plane. A similar characterization based on $L^p$-infinite-time admissibility is provided for multiplication semigroups on $L^q$-spaces with $1 \leq q \leq p < \infty$. For polynomially stable $C_0$-semigroups on Hilbert spaces, we also give a sufficient condition for $L^2$-infinite-time admissibility.