  We demonstrate that the structure of complex second-order strongly elliptic
operators $H$ on ${\bf R}^d$ with coefficients invariant under translation by
${\bf Z}^d$ can be analyzed through decomposition in terms of versions $H_z$,
$z\in{\bf T}^d$, of $H$ with $z$-periodic boundary conditions acting on
$L_2({\bf I}^d)$ where ${\bf I}=[0,1>$. If the semigroup $S$ generated by $H$
has a H\"older continuous integral kernel satisfying Gaussian bounds then the
semigroups $S^z$ generated by the $H_z$ have kernels with similar properties
and $z\mapsto S^z$ extends to a function on ${\bf C}^d\setminus\{0\}$ which is
analytic with respect to the trace norm. The sequence of semigroups $S^{(m),z}$
obtained by rescaling the coefficients of $H_z$ by $c(x)\to c(mx)$ converges in
trace norm to the semigroup $\hat{S}^z$ generated by the homogenization
$\hat{H}_z$ of $H_z$. These convergence properties allow asymptotic analysis of
the spectrum of $H$.
