On estimates of the approximation numbers of the Hardy operator

We obtain two–sided estimates which describe the behaviour of the approximation numbers of the Hardy operator and Schatten–Neumann norms in the new case, when the compact operator
$$ Tf(x)=\int_0^x f(\tau) d\tau, \quad x>0, $$
is acting from a Lebesgue space to a Lorentz space $(T: L_v^r(R^+)\to L_\omega^{pq}(R^+))$ under the condition $1<p<r\leqslant q<\infty$.