  Let $c_{kl} \in W^{2,\infty}(\mathbb{R}^d, \mathbb{C})$ for all $k,l \in \{1,
\ldots, d\}$. We consider the divergence form operator $A = - \sum_{k,l=1}^d
\partial_l (c_{kl} \, \partial_k) $in $L_2(\mathbb{R}^d)$ when the coefficient
matrix satisfies $(C(x) \, \xi, \xi) \in \Sigma_\theta$ for all $x \in
\mathbb{R}^d$ and $\xi \in \mathbb{C}^d$, where $\Sigma_\theta$ be the sector
with vertex 0 and semi-angle $\theta$ in the complex plane. We show that for
all $p$ in a suitable interval the contraction semigroup generated by $-A$
extends consistently to a contraction semigroup on $L_p(\mathbb{R}^d)$. For
those values of $p$ we present a condition on the coefficients such that the
space $C_c^\infty(\mathbb{R}^d)$ of test functions is a core for the generator
on $L_p(\mathbb{R}^d)$. We also examine the operator $A$ separately in the more
special Hilbert space $L_2(\mathbb{R}^d)$ setting and provide more sufficient
conditions such that $C_c^\infty(\mathbb{R}^d)$ is a core.
