Numerical Range and Quasi-Sectorial Contractions

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 $\{\exp (-tS){\}}_{t\ge 0}$, 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: $\exp (-tS)=\underset{n\to \infty }{\lim }(I+tS/n{)}^{-n}$, $t\ge 0$, for this class of semigroups.