The growth of a C_0-semigroup characterised by its cogenerator

TL;DR

This paper characterizes key properties of C_0-semigroups, such as contractivity and polynomial boundedness, through their cogenerators, extending existing results and providing sharp examples across different Banach spaces.

Contribution

It extends the characterization of semigroup properties via cogenerators to broader settings and establishes the optimality of these results with concrete examples.

Findings

Polynomial boundedness of a semigroup implies polynomial boundedness of its cogenerator.

For analytic semigroups, the converse of polynomial boundedness holds.

Examples show the sharpness of the characterizations and the failure of the Foias-Sz.-Nagy theorem outside Hilbert spaces.

Abstract

We characterise contractivity, boundedness and polynomial boundedness for a C_0-semigroup on a Banach space in terms of its cogenerator V (or the Cayley transform of the generator) or its resolvent. In particular, we extend results of Gomilko and Brenner, Thomee and show that polynomial boundedness of a semigroup implies polynomial boundedness of its cogenerator. As is shown by an example, the result is optimal. For analytic semigroups we show that the converse holds, i.e., polynomial boundedness of the cogenerators implies polynomial boundedness of the semigroup. In addition, we show by simple examples in (C^2,\|\cdot\|_p), p \neq 2, that our results on the characterisation of contractivity are sharp. These examples also show that the famous Foias-Sz.-Nagy theorem on cogenerators of contraction semigroups on Hilbert spaces fails in (C^2,\|\cdot\|_p) for p\neq 2.