Analyticity of the Dirichlet-to-Neumann semigroup on continuous functions

Let $\mathrm{\Omega }$ be a bounded open subset with $C^{1+\kappa }$-boundary for some $\kappa >0$. Consider the Dirichlet-to-Neumann operator associated to the elliptic operator $-\sum {\mathrm{\partial }}_l(c_{kl}{\mathrm{\partial }}_k)+V$, where the $c_{kl}=c_{lk}$ are H"older continuous and $V\in L_{\mathrm{\infty }}(\mathrm{\Omega })$ are real valued. We prove that the Dirichlet-to-Neumann operator generates a $C_0$-semigroup on the space $C(\mathrm{\partial }\mathrm{\Omega })$ which is in addition holomorphic with angle $\frac{\pi }{2}$. We also show that the kernel of the semigroup has Poisson bounds on the complex right half-plane. As a consequence we obtain an optimal holomorphic functional calculus and maximal regularity on $L_p(\mathrm{\Gamma })$ for all $p\in (1,\mathrm{\infty })$.