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arXiv · 1506.05974

Catalysis in the trace class and weak trace class ideals

Guillaume Aubrun·Fedor Sukochev·Dmitriy Zanin
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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.

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Guillaume Aubrun, Fedor Sukochev, Dmitriy Zanin. 2015-06-19. Catalysis in the trace class and weak trace class ideals. https://arxiv.org/abs/1506.05974

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