Factorization of operator valued Lp for 0≤p<1

In [5], it is proved that a bounded linear operator u, from a Banach space Y into an L,(S, v) factors through L-p1 (S, nu) for some p(1) > 1, if Y* is of finite cotype; (S, nu) is a probability space for p = 0, and any measure space for 0 < p < 1. In this paper, we generalize this result to uupsilon, where u : Y --> L-p(S, nu) and upsilon : X --> Y are linear operators such that upsilon* is of finite Kasin cotype. This result gives also a new proof of Grothendieck's theorem. (C) 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.