Notes on the Chernoff estimate

TL;DR

This paper refines the Chernoff estimate for contraction semigroups on Banach spaces, extending it from strong to operator-norm topology, and applies it to prove convergence of Dunford-Segal approximants.

Contribution

It introduces a modified $ oot{n} ext{-estimate}$ for contractions and extends Chernoff estimates from strong to operator-norm topology for quasi-sectorial semigroups.

Findings

Modified $ oot{n} ext{-estimate}$ for contractions on Banach spaces.

Operator-norm Chernoff estimate for quasi-sectorial contraction semigroups.

Proved convergence of Dunford-Segal approximants in operator norm.

Abstract

The purpose of the present notes is to examine the following issues related to the the Chernoff estimate: (1) For contractions on a Banach space we modify the $\sqrt{n}$-estimate and apply it in the proof of the Chernoff product formula for $C_0$-semigroups in the \textit{strong} operator topology. (2) We use the idea of a {probabilistic} approach, proving the Chernoff estimate in the strong operator topology, to uplift it to the \textit{operator-norm} estimate for \textit{quasi-sectorial} contraction semigroups. (3) The operator-norm Chernoff estimate is applied to {quasi-sectorial} contraction semigroups for proving the operator-norm convergence of the \textit{Dunford-Segal} approximants.