Numerical Range and Quasi-Sectorial Contractions

TL;DR

This paper investigates the numerical range of quasi-sectorial contraction semigroups, establishing a relation with their generators and providing a new proof for the convergence rate of the Euler approximation method.

Contribution

It introduces a novel relation between the numerical range of quasi-sectorial contraction semigroups and their sectorial generators, along with a new proof for the Euler formula approximation rate.

Findings

Established a relation between numerical range and generators of quasi-sectorial contraction semigroups

Provided a new proof of the $O(1/n)$ convergence rate for the Euler approximation

Applied a method to localize the numerical range of these semigroups

Abstract

We apply a method developed by one of the authors, see \cite{Arl1}, to localize the numerical range of \textit{quasi-sectorial} contractions semigroups. Our main theorem establishes a relation between the numerical range of quasi-sectorial contraction semigroups $\{\exp(- t S)\}_{t\ge 0}$, and the maximal {sectorial} generators $S$. We also give a new prove of the rate $O(1/n)$ for the operator-norm Euler formula approximation: $\exp(- t S)=\lim\limits_{n\to \infty}(I+tS/n)^{-n}$, $t\ge 0$, for this class of semigroups.