Universal Non-Completely-Continuous Operators

Abstract

A bounded linear operator between Banach spaces is called {\it completely continuous} if it carries weakly convergent sequences into norm convergent sequences. Isolated is a universal operator for the class of non-completely-continuous operators from $L_1$ into an arbitrary Banach space, namely, the operator from $L_1$ into ${\ell }_{\infty }$ defined by $$T_0(f)={\left(\int r_{n}fd\mu \right)}_{n\ge 0},$$ where $r_n$ is the $n^{\text{th}}$ Rademacher function. It is also shown that there does not exist a universal operator for the class of non-completely-continuous operators between two arbitrary Banach space. The proof uses the factorization theorem for weakly compact operators and a Tsirelson-like space.