Criteria in terms of weights v and w are given for the compactness of the two–dimensional Hardy operator I_2 from Lebesgue space L^p_v (R^2_+) to L^q_w(R^2_+) for 1 < p q < ∞. A two–sided estimate is found for the measure of non–compactness of I_2 : L^p_v (R^2_+) → L^q_w(R^2_+) for the same case of summation parameters p and q. The situation when q < p is also discussed.