Optimal polynomial decay of functions and operator semigroups

TL;DR

This paper characterizes the polynomial decay rates of operator semigroups in Hilbert and Banach spaces using resolvent conditions, confirming the sharpness of recent results and settling a conjecture in the field.

Contribution

It provides a resolvent-based characterization of polynomial decay for semigroups and proves the sharpness of these results in Banach spaces, settling a prior conjecture.

Findings

Polynomial decay characterized by resolvent conditions.

Results are sharp for Banach space semigroups.

Settles a conjecture on decay rates in the literature.

Abstract

We characterize the polynomial decay of orbits of Hilbert space $C_0$-semigroups in resolvent terms. We also show that results of the same type for general Banach space semigroups and functions obtained recently in the paper by C.J.K.Batty and T.Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces (J. Evol. Eq. 2008), are sharp. This settles a conjecture posed in the paper by C.J.K.Batty and T.Duyckaerts.