Let Γ ⊂ Rd be a lattice. For ϵ > 0, we consider the perforated space " ⊂ Rd which is an ("Γ)-periodic open connected set with Lipschitz boundary. In L2(IIϵ;Cn), we consider a self-adjoint strongly elliptic second-order differential operator Aϵ with periodic coefficients depending on x=ϵ. We study the behavior of the resolvent (Aϵ + I)-1 for small ϵ. Approximations for this resolvent in the (L2 → L2) and (L2 → H1)-operator norms with sharp order error estimates are found. The results are obtained by the operator-theoretic (spectral) approach. General results are applied to particular periodic operators of mathematical physics: the acoustics operator, the elasticity operator, and the Schrödinger operator with a singular potential.