  Let $\mathbb K(\mathbb R^d)$ denote the cone of discrete Radon measures on
$\mathbb R^d$. The gamma measure $\mathcal G$ is the probability measure on
$\mathbb K(\mathbb R^d)$ which is a measure-valued L\'evy process with
intensity measure $s^{-1}e^{-s}\,ds$ on $(0,\infty)$. We study a class of
Laplace-type operators in $L^2(\mathbb K(\mathbb R^d),\mathcal G)$. These
operators are defined as generators of certain (local) Dirichlet forms. The
main result of the papers is the essential self-adjointness of these operators
on a set of `test' cylinder functions on $\mathbb K(\mathbb R^d)$.
