Universally symmetric norming operators are compact

Abstract

We introduce and study the concepts of "universally symmetrically norming operators" and "universally absolutely symmetrically norming operators" on a separable Hilbert space. These refer to the operators that are, respectively, norming and absolutely norming, with respect to every symmetric norm on B(H). We establish a characterization theorem for such operators and prove that these classes are identical and that they coincide with the class of compact operators. In particular, we provide an alternative characterization of compact operators on a separable Hilbert space.