  Given an elliptic differential operator L of second order with smooth
coefficients in a bounded domain with smooth boundary. We show that if the
coefficients are H\"older-continuous up to the boundary and the boundary is
$C^{2,\alpha}$-smooth that on the space of all $C^{2,\alpha}$-smooth (up to the
boundary) functions u fulfilling both u=0 and Lu=0 (on the boundary) the
operator L is dissipative and closable to an generator of a strong continuous
operator semigroup in the space of continuous functions with zero boundary
condition. Moreover we show that if the coefficients of the second order and
first order derivatives are in $C^{2,\alpha}$ then the above mentioned subspace
of $C^{2,\alpha}$ which is a core for the generator of the semigroup, can be
embedded (continued) in a contractive and smooth way. Thus we construct a
linear extension operator which maps a $C^{2,\alpha}$-smooth function
fulfilling the boundary conditions to a $C^{2,\alpha}$-smooth function in
$\R^n$ such that the supremum is not increased.
