Weakly compact operators onH ?

Abstract

We prove that a weakly compact operator fromH∞ or any of its even duals into an arbitrary Banach space is uniformly convexifying. By using this, we establish three dicothomies: (1) every operator defined onH∞ or any of its even duals either fixes a copy ofl∞ or factors through a Banach space having the Banach-Saks property; (2) every quotient ofH∞ or any of its even duals either contains a copy ofl∞ or is super-reflexive; (3) every subspace ofL1/H 0 1 or any of its even duals either contains a complemented copy ofl1 or is super-reflexive.