Catalysis in the trace class and weak trace class ideals

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 $\textrm {Tr} (B^p) \leq \textrm {Tr} (A^p)$ 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.