# Von Neumann Type of Trace Inequalities for Schatten-Class Operators

## TL;DR

This paper extends von Neumann's trace inequality and eigenvalue inequalities from finite-dimensional matrices to Schatten-class operators on infinite-dimensional Hilbert spaces, utilizing recent results on the $C$-numerical range.

## Contribution

It introduces a generalization of classical trace and eigenvalue inequalities to Schatten-class operators in infinite-dimensional settings, expanding their applicability.

## Key findings

- Generalized von Neumann's trace inequality to Schatten-class operators.
- Extended eigenvalue inequalities for hermitian operators.
- Utilized recent $C$-numerical range results for the generalization.

## Abstract

We generalize von Neumann's well-known trace inequality, as well as related eigenvalue inequalities for hermitian matrices, to Schatten-class operators between complex Hilbert spaces of infinite dimension. To this end, we exploit some recent results on the $C$-numerical range of Schatten-class operators. For the readers' convenience, we sketched the proof of these results in the Appendix.