On the numerical range of generators of symmetric L_infty-contractive semigroups

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.