Let $H$ be the generator of a symmetric Markov semigroup on $L^2(\mathbb{R}^n).$ Why the fractional power $H^\alpha$ (defined on a proper domain) with $0 < \alpha < 1$ turn out to be the generator of another symmetric Markov semigroup?
I am trying to use the integral formula
$$ s^\alpha = {\sin(\pi \alpha) \over \pi} \int_{0}^\infty t^{\alpha-1} {s \over t+s} \: dt \quad \mbox{for all } 0 < \alpha < 1. $$
Do you have any hint?