On the entropy of absolutely summing operators (Q1063845)

Let E, F be Banach spaces. For a linear operator S:\(E\to F\) let \(e_ n(S)\) denote the n-th entropy number of S. \(\ell_{a,b}\), \(0<a,b\leq \infty\) denotes the Lorentz sequence space. It is proved that if E' and F' are type 2 spaces, then the condition \(e_ n(S)\in \ell_{2,1}\) implies that S is an r-summing operator and if S is an r-summing operator, then \((e_ n(S))\in \ell_{2,\infty}\) \((0<r<\infty)\).