Relation between Quasi-normed Ideals of Entropy Numbers and of Approximation Numbers

Abstract

Let en(T) and an(T) be the entropy numbers and approximation numbers of operator T between Banach spaces, respectively. Let ?p be the operator ideal defined by Sp:^TeL;{en(T)^lp}, and Sp be the operator ideal defined by Sp:=|TeL;£an(T)P<°^. Then if 0<p<l, and 1 <u, v<°° we have Sp(lu, lv)&?p0u, lv).