The number of elements required to determine \((p,1)\)-summing norms (Q1892738)

The author proves that for all operators \(T\) of rank \(n\) between normed spaces, the (2,1)-summing norm of \(T\) can be calculated essentially by \(n \ln n\) elements, i.e. with some fixed constant \(c\) (and \(n \geq 3\)) \[ \pi_{2,1} (T) \leq c \pi_{2,1}^{(n \ln n)} (T). \] \vskip3mm For \(p > 2\), the argument yields a simplified proof of the fact that \[ \pi_{p, 1} (T) \leq c_p \pi^{(n)}_{p,1} (T), \] where \(c_p = O((p - 2)^{-1/2})\). The method uses estimates for the number of vectors required in terms of the quotient \(\pi_1 (T) /\pi_{p,1} (T)\).