Catalysis in the trace class and weak trace class ideals

2015/06/19 by Guillaume Aubrun, Fedor Sukochev, Aubrun, Guillaume +3
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #math.FA

paper · pdf · doi:10.48550/arxiv.1506.05974

arxiv created 2015/06/19 · arxiv updated 2015/06/22

Abstract

Given operators A,B in some ideal I in the algebra 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 ⊗ C is submajorized (in the sense of Hardy--Littlewood) by A ⊗ C ? In the case when I = L₁, a necessary and almost sufficient condition is that the inequalities \rm Tr (B^p) ≤ \rm Tr (A^p) hold for every p ∈ [1,∞]. We show that the analogous statement fails for I = L_1,∞ by connecting it with the study of Dixmier traces.

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