The paper studies a bounded symmetric operator Aε in
L2(Rd) with (Aεu)(x)=ε−d−2∫Rda((x−y)/ε)μ(x/ε,y/ε)(u(x)−u(y))dy; here ε is a small
positive parameter. It is assumed that a(x) is a non-negative
L1(Rd) function such that a(−x)=a(x) and the moments Mk=∫Rd∣x∣ka(x)dx, k=1,2,3, are finite. It is also assumed
that μ(x,y) is Zd-periodic both in x and y function such
that μ(x,y)=μ(y,x) and 0<μ−≤μ(x,y)≤μ+<∞. Our
goal is to study the limit behaviour of the resolvent
(Aε+I)−1, as ε→0. We show that, as
ε→0, the operator (Aε+I)−1
converges in the operator norm in L2(Rd) to the resolvent
(A0+I)−1 of the effective operator A0 being a
second order elliptic differential operator with constant coefficients of the
form A0=−divg0∇. We then obtain sharp in
order estimates of the rate of convergence