Universal non-compact operators between super-reflexive Banach spaces and the existence of a complemented copy of Hilbert space

Abstract

Suppose that 1<p≦2, 2≦q<∞. The formal identity operatorI:lp→lqfactorizes through any given non-compact operator from ap-smooth Banach space into aq-convex Banach space. It follows that ifX is a 2-convex space andY is an infinite dimensional subspace ofX which is isomorphic to a Hilbert space, thenY contains an isomorphic copy ofl2 which is complemented inX.