  We present an operator-asymptotic approach to the problem of homogenization
of periodic composite media in the setting of three-dimensional linearized
elasticity. This is based on a uniform approximation with respect to the
inverse wavelength $|\chi|$ for the solution to the resolvent problem when
written as a superposition of elementary plane waves with wave vector
(``quasimomentum") $\chi$. We develop an asymptotic procedure in powers of
$|\chi|$, combined with a new uniform version of the classical Korn inequality.
As a consequence, we obtain $L^2\to L^2$, $L^2\to H^1$, and higher-order
$L^2\to L^2$ norm-resolvent estimates in $\mathbb{R}^3$. The $L^2 \to H^1$ and
higher-order $L^2 \to L^2$ correctors emerge naturally from the asymptotic
procedure, and the former is shown to coincide with the classical formulae.
