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International Journal of Theoretical Physics

, Volume 56, Issue 12, pp 3819–3830 | Cite as

On τ-Compactness of Products of τ-Measurable Operators

Article
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Abstract

Let \(\mathcal {M}\) be a von Neumann algebra of operators on a Hilbert space \(\mathcal {H}\), τ be a faithful normal semifinite trace on \(\mathcal {M}\). We obtain some new inequalities for rearrangements of τ-measurable operators products. We also establish some sufficient τ-compactness conditions for products of selfadjoint τ-measurable operators. Next we obtain a τ-compactness criterion for product of a nonnegative τ-measurable operator with an arbitrary τ-measurable operator. We construct an example that shows importance of nonnegativity for one of the factors. The similar results are obtained also for elementary operators from \(\mathcal {M}\). We apply our results to symmetric spaces on \((\mathcal {M}, \tau )\). The results are new even for the *-algebra \(\mathcal {B}(\mathcal {H})\) of all linear bounded operators on \(\mathcal {H}\) endowed with the canonical trace τ = tr.

Keywords

Hilbert space Linear operator Von Neumann algebra Normal semifinite trace τ-measurable operator τ-compact operator Elementary operator Integrable operator Rearrangement 

Notes

Acknowledgments

The author is grateful to the Referee for helpful remarks.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Kazan (Volga region) Federal UniversityKazanRussia

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