Gaussian estimates and holomorphy of semigroups

We show that if a selfadjoint semigroup T on ${L^2}(\Omega )$ satisfies a Gaussian estimate $\vert T(t)f\vert \leq MG(bt)\vert f\vert,0 \leq t \leq 1,f \in {L^2}(\Omega )$ (where $G = G{(t)_{t \geq 0}}$ is the Gaussian semigroup on ${L^2}({R^N})$ and $\Omega$ is an open set of ${R^N}$), then T defines a holomorphic semigroup of angle $\frac{\pi }{2}$ on ${L^p}(\Omega )$ . We obtain by duality the same result on ${C_0}(\Omega )$. Applications to uniformly elliptic operators and Schrödinger operators are given.