could anyone tell me what a $C^\infty$ regularizing contraction semigroup is? I know the contraction semigroup part, but I don't understand what exactly the $C^\infty$ regularizing part.
De regularizing semigroups mean $e^{-tL}u_0$ is more regular if $u_0$ belongs to a more regular subset of the domain of $L$? I looked it up by myself, but I didn't find any proper definition.
I'd appreciate it if you'd help me. Thank you.