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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 coefficientmatrix 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 sectorwith vertex 0 and semi-angle $\theta$ in the complex plane. We show that forall $p$ in a suitable interval the contraction semigroup generated by $-A$extends consistently to a contraction semigroup on $L_p(\mathbb{R}^d)$. Forthose values of $p$ we present a condition on the coefficients such that thespace $C_c^\infty(\mathbb{R}^d)$ of test functions is a core for the generatoron $L_p(\mathbb{R}^d)$. We also examine the operator $A$ separately in the morespecial Hilbert space $L_2(\mathbb{R}^d)$ setting and provide more sufficientconditions such that $C_c^\infty(\mathbb{R}^d)$ is a core.

Degenerate elliptic operators in $L_p$-spaces with complex $W^{2,\infty}$-coefficients