  We prove two assumptions made in an article by Ya.A. Butko, M. Grothaus, O.G.
Smolyanov concerning the existence of a strongly continuous operator semigroup
solving a Cauchy-Dirichlet problem for an elliptic differential operator in a
bounded domain and the existence of a smooth contractive embedding of a core of
the generator of the semigroup into the space $C_c^{2,\alpha}(\R^n)$. Based on
these assumptions a Feynman formula for the solution of the Cauchy-Dirichlet
problem is constructed in the article mentioned above. In this article we show
that the assumptions are fulfilled for domains with $C^{4,\alpha}$-smooth
boundary and coefficients in $C^{2,\alpha}$.
