Catalysis in the trace class and weak trace class ideals
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- by Guillaume Aubrun, Fedor Sukochev and Dmitriy Zanin
- Proc. Amer. Math. Soc. 144 (2016), 2461-2471
- DOI: https://doi.org/10.1090/proc/12889
- Published electronically: October 20, 2015
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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 $\textrm {Tr} (B^p) \leq \textrm {Tr} (A^p)$ 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.References
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Bibliographic Information
- Guillaume Aubrun
- Affiliation: Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France
- Email: aubrun@math.univ-lyon1.fr
- Fedor Sukochev
- Affiliation: School of Mathematics and Statistics, University of NSW, Sydney, 2052, Australia
- MR Author ID: 229620
- Email: f.sukochev@unsw.edu.au
- Dmitriy Zanin
- Affiliation: School of Mathematics and Statistics, University of NSW, Sydney, 2052, Australia
- MR Author ID: 752894
- Email: d.zanin@unsw.edu.au
- Received by editor(s): March 25, 2015
- Received by editor(s) in revised form: June 18, 2015, and July 3, 2015
- Published electronically: October 20, 2015
- Additional Notes: The research of the first author was supported by the ANR projects OSQPI (ANR-11-BS01-0008) and StoQ (ANR-14-CE25-0003)
The research of the second and third authors has been supported by the ARC projects DP140100906 and DP 120103263. - Communicated by: Pamela B. Gorkin
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 2461-2471
- MSC (2010): Primary 47A80, 47B10, 47L20
- DOI: https://doi.org/10.1090/proc/12889
- MathSciNet review: 3477062