We continue to study a class of matrix periodic elliptic second order differential operators ${{\mathcal A}_\epsilon}$ in ${\mathbb{R}^d}$ with rapidly oscillating coefficients (depending on ${\mathbf{x}/\epsilon}$). This class was considered in earlier papers by the authors. The homogenization problem in the small period limit is studied. We obtain approximation for the resolvent ${({\mathcal A}_\epsilon + I)^{-1}}$ in the operator norm from ${L_2(\mathbb{R}^d)}$ to ${H^1(\mathbb{R}^d)}$ with error of order ${\epsilon}$. In this approximation, the corrector is taken into accout. Besides, the ( ${L_2} \to {L_2}$) -approximations of the so called fluxes are obtained.