A Note on Universal Operators

Abstract

An operator T is called universal for the complement of the ideal A if T does not belong to A, and factors through every element of the complement of A. We show that the complements of many ideals (such as the ideal of strictly (co)singular operators, or any maximal normed ideal) have no universal operators. On the other hand, the complement of the ideal of finitely strictly singular operators has a universal operator. Moreover, we show that, for many ideals A, any positive operator which factors positively through any positive member of the complement of A must be compact.