Functional calculus for generators of symmetric contraction semigroups

We prove that every generator of a symmetric contraction semigroup on a $\sigma$-finite measure space admits, for $1<p<\mathrm{\infty }$, a H"ormander-type holomorphic functional calculus on $L^p$ in the sector of angle ${\varphi }_p^{\ast }=\arcsin \mathrm{\mid }1-2\mathrm{/}p\mathrm{\mid }$. The obtained angle is optimal.