How many vectors are needed to compute \((p,q)\)-summing norms (Q1913500)

If \(X\) and \(Y\) are Banach spaces, \(T\) is in \({\mathcal L}(X,Y)\), and \(1\leq q\leq p\leq +\infty\), then for any integer \(n\geq 1\) let \[ \Pi^n_{pq}(T)= \sup|\{Tx_i\}^n_{i=1}|_p, \] where the supremum is taken over all \(\{x_i\}^n_{i=1}\) in \(X\) for which \[ \sup_{|f|=1} |\{\langle f,x_i\rangle\}^n_{i=1}|_q\leq 1. \] If \(\Pi_{pq}(T)= \sup_n \Pi^n_{pq}(T)<+\infty\), then \(T\) is called absolutely \(pq\)-summing. A problem of Figiel is: Given \(p\), \(q\) (as above) and some \(n\geq 1\), what is the smallest constant \(k_n\) so that for some constant \(c\) we have \[ \Pi_{pq}(T)\leq c \Pi^{k_n}_{pq}(T) \] for all operators \(T\) in \({\mathcal L}(X,Y)\) of rank \(n\)? The current paper contains a number of results which extend and complement previous work of several other authors on this problem.