An inequality between thep- and (p, 1)-summing norm of finite rank operators fromC(K)-spaces

We show, for any operatorT from aC(K)-space into a Banach space with rank (T)≤n, the inequality $$\pi _p (T) \leqslant C(1 + log n)^{1 - 1/p} \pi _{p,1} (T), 1< p< \infty $$ , whereC≤4.671 is a numerical constant. The factor (1+logn)1−1/p is asymptotically correct. This inequality extends a result of Jameson top ≠ 2. Several applications are given — one is a positive solution of a conjecture of Rosenthal and Szarek: For 1≤p