# Universally symmetric norming operators are compact

## TL;DR

This paper characterizes compact operators on infinite-dimensional Hilbert spaces as those that are universally norming across all symmetric norms, extending previous results and providing new insights into operator theory.

## Contribution

It introduces the concepts of universally symmetric norming and absolutely symmetric norming operators, proving they coincide and are exactly the compact operators.

## Key findings

- Universal symmetric norming operators are exactly the compact operators.
- The class of universally symmetric norming and absolutely symmetric norming operators are identical.
- Provides an alternative characterization of compact operators on separable infinite-dimensional Hilbert spaces.

## Abstract

We study a specific family of symmetric norms on the algebra $\mathcal B(\mathcal H)$ of operators on a separable infinite-dimensional Hilbert space. With respect to each symmetric norm in this family the identity operator fails to attain its norm. Using this, we generalize one of the main results from \cite{SP}; the hypothesis is relaxed, and consequently, the family of symmetric norms for which the result holds is extended. We introduce and study the concepts of "universally symmetric norming operators" and "universally absolutely symmetric 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 $\mathcal B(\mathcal H)$. 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 infinite-dimensional Hilbert space.