A remark on the growth rate of operator semigroups under resolvent bounds

TL;DR

This paper establishes a new growth bound for operator semigroups on Hilbert spaces based on resolvent bounds, offering sharper estimates for certain super-linear growth scenarios compared to existing results.

Contribution

It introduces improved growth estimates for $C_0$-semigroups under resolvent bounds, enhancing the understanding of their behavior in Hilbert spaces.

Findings

New growth bounds for operator semigroups derived

Sharper estimates for super-linear resolvent growths

Enhanced understanding of semigroup behavior under resolvent constraints

Abstract

We provide a growth bound for the operator norm of $C_0$-semigroups on Hilbert spaces under a corresponding growth bound on the resolvent of the semigroup generator. For some super-linear resolvent growths, our estimate is sharper than the ones currently available in the literature.