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Two classes of τ-measurable operators affiliated with a von Neumann algebra

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Abstract

Let M be a von Neumann algebra of operators on a Hilbert space H, τ be a faithful normal semifinite trace on M. We define two (closed in the topology of convergence in measure τ) classes P 1 and P 2 of τ-measurable operators and investigate their properties. The class P 2 contains P 1. If a τ-measurable operator T is hyponormal, then T lies in P 1; if an operator T lies in P k , then UTU* belongs to P k for all isometries U from M and k = 1, 2; if an operator T from P 1 admits the bounded inverse T −1, then T −1 lies in P 1. We establish some new inequalities for rearrangements of operators from P 1. If a τ-measurable operator T is hyponormal and T n is τ-compact for some natural number n, then T is both normal and τ-compact. If M = B(H) and τ = tr, then the class P 1 coincides with the set of all paranormal operators on H.

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  1. Kazan (Volga Region) Federal University, ul. Kremlyovskaya 18, Kazan, 420008, Russia

    A. M. Bikchentaev

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  1. A. M. Bikchentaev
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Correspondence to A. M. Bikchentaev.

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Original Russian Text © A.M. Bikchentaev, 2017, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2017, No. 1, pp. 86–91.

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Bikchentaev, A.M. Two classes of τ-measurable operators affiliated with a von Neumann algebra. Russ Math. 61, 76–80 (2017). https://doi.org/10.3103/S1066369X17010091

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