On the numerical range of generators of symmetric $L_\infty$-contractive semigroups

TL;DR

This paper provides an elementary proof that the optimal angle of analyticity for symmetric submarkovian semigroups on Lp spaces remains valid even without the positivity assumption, extending previous results.

Contribution

It offers a simplified proof of Kriegler's 2011 result on the analyticity angle of symmetric semigroups without positivity assumptions.

Findings

The optimal angle of analyticity is preserved without positivity.

Elementary proof simplifies understanding of semigroup analyticity.

Extends previous results to broader class of semigroups.

Abstract

A result by Liskevich and Perelmuter from 1995 yields the optimal angle of analyticity for symmetric submarkovian semigroups on $L_p$, $1<p<\infty$. C.~Kriegler showed in 2011 that the result remains true without the assumption of positivity of the semigroup. Here we give an elementary proof of Kriegler's result.