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. 
Markus Haase, Peer Christian Kunstmann, Hendrik Vogt
math.FA