The Dirichlet-to-Neumann operator on $C(\mathrm{\partial }\mathrm{\Omega })$

Let $\mathrm{\Omega }\subset {\mathbf{R}}^d$ be an open bounded set with Lipschitz boundary $\mathrm{\Gamma }$. Let $D_V$ be the Dirichlet-to-Neumann operator with respect to a purely second-order symmetric divergence form operator with real Lipschitz continuous coefficients and a positive potential $V$. We show that the semigroup generated by $-D_V$ leaves $C(\mathrm{\Gamma })$ invariant and that the restriction of this semigroup to $C(\mathrm{\Gamma })$ is a $C_0$-semigroup. We investigate positivity and spectral properties of this semigroup. We also present results where $V$ is allowed to be negative. Of independent interest is a new criterium for semigroups to have a continuous kernel.