Numerical Range and Quasi-Sectorial Contractions

Yury Arlinskii (1) , Valentin A. Zagrebnov (2, 3)

1 Department of Mathematical Analysis
2 CPT - Centre de Physique Théorique - UMR 6207
3 CPT - Centre de Physique Théorique - UMR 7332

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.

Mots clés: semigroups on the complex plane; Operator numerical range; maximal sectorial generators; quasi-sectorial contractions

Domaines: Analyse fonctionnelle [math.FA]

Yury Arlinskii, Valentin A. Zagrebnov. Numerical Range and Quasi-Sectorial Contractions. Journal of Mathematical Analysis and Applications, 2010, 366 (1), pp.33-43. ⟨10.1016/j.jmaa.2010.01.036⟩. ⟨hal-00331492⟩