  We study the asymptotic behaviour of the resolvents $({\mathcal
A}^\varepsilon+I)^{-1}$ of elliptic second-order differential operators
${\mathcal A}^\varepsilon$ in ${\mathbb R}^d$ with periodic rapidly oscillating
coefficients, as the period $\varepsilon$ goes to zero. The class of operators
covered by our analysis includes both the "classical" case of uniformly
elliptic families (where the ellipticity constant does not depend on
$\varepsilon$) and the "double-porosity" case of coefficients that take
contrasting values of order one and of order $\varepsilon^2$ in different parts
of the period cell. We provide a construction for the leading order term of the
"operator asymptotics" of $({\mathcal A}^\varepsilon+I)^{-1}$ in the sense of
operator-norm convergence and prove order $O(\varepsilon)$ remainder estimates.
