Catalysis in the trace class and weak trace class ideals

TL;DR

This paper investigates conditions under which operators in trace and weak trace ideals can be related through submajorization, revealing that certain inequalities suffice in trace class but not in weak trace class, linked to Dixmier traces.

Contribution

It establishes that the trace inequalities characterize submajorization in trace class operators but fail in weak trace class, highlighting differences between these operator ideals.

Findings

Trace inequalities are necessary and almost sufficient for submajorization in trace class.

The analogous conditions do not hold in weak trace class, demonstrated via Dixmier traces.

The study clarifies the limitations of trace inequalities in weak trace class operator comparisons.

Abstract

Given operators $A,B$ in some ideal $\mathcal{I}$ in the algebra $\mathcal{L}(H)$ of all bounded operators on a separable Hilbert space $H$, can we give conditions guaranteeing the existence of a trace-class operator $C$ such that $B \otimes C$ is submajorized (in the sense of Hardy--Littlewood) by $A \otimes C$ ? In the case when $\mathcal{I} = \mathcal{L}_1$, a necessary and almost sufficient condition is that the inequalities ${\rm Tr} (B^p) \leq {\rm Tr} (A^p)$ hold for every $p \in [1,\infty]$. We show that the analogous statement fails for $\mathcal{I} = \mathcal{L}_{1,\infty}$ by connecting it with the study of Dixmier traces.