On $L^p$-contractivity of Laguerre semigroups

We study several Laguerre semigroups appearing in the literature and find sharp ranges of type parameters for which these semigroups are contractive on all $L^p$ spaces, $1\le p\le \mathrm{\infty }$. We also answer a similar question for Bessel semigroups, which in a sense are closely related to the Laguerre semigroups.