The paper deals with homogenization of self-adjoint operators in $L_2(\mathbb R^d)$ of the form $$ ({ \mathbb A}_\eps u) (\x) = \int_{ \R^d} \mu(\x/\eps, \y/\eps) \frac{ \left( u(\x) - u(\y) \right)}{ |\x - \y|^{ d+\alpha}}\,d\y, $$ where $0< \alpha < 2$, and $\eps>0$ is a small parameter. It is assumed that the function $\mu(\x,\y)$ is $\Z^d$-periodic in each variable, $\mu(\x,\y)=\mu(\y,\x)$ for all $\x$ and $\y$, and $0< \mu_- \leqslant \mu(\x,\y) \leqslant \mu_+< \infty$. Under these assumptions we show that the resolvent $({ \mathbb A}_\eps + I)^{ -1}$ converges, as $\eps\to0$, in the operator norm in $L_2(\R^d)$ to the resolvent $({ \mathbb A}^0 + I)^{ -1}$ of the limit operator ${ \mathbb A}^0$ given by $$ ({ \mathbb A}^0 u) (\x) = \int_{ \R^d} \mu^0 \frac{ \left( u(\x) - u(\y) \right)}{ |\x - \y|^{ d+\alpha}}\,d\y, $$ where $\mu^0$ is the mean value of $\mu(\x,\y)$. We also show that the operator norm of the discrepancy $\|({ \mathbb A}_\eps + I)^{ -1} - (\A^0 + I)^{ -1}\|_{ L_2(\mathbb R^d)\to L_2(\mathbb R^d)}$ can be estimated by $O(\eps^\alpha)$, if $0< \alpha < 1$, by $O(\eps (1 + | \operatorname{ ln} \eps|)^2)$, if $ \alpha =1$, and by $O(\eps^{ 2- \alpha})$, if $1< \alpha < 2$.
Categories: math.AP math.FA