On an analog of the M.G. Krein theorem for measurable operators

A.M. Bikchentaev

Kazan Federal University, Kazan, 420008 Russia

E-mail: Airat.Bikchentaev@kpfu.ru

Received October 12, 2017

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Abstract

Let M be a von Neumann algebra of operators on a Hilbert space H and τ be a faithful normal semifinite trace on M. Let μ_t(T), t > 0, be a rearrangement of a τ-measurable operator T. Let us consider a τ-measurable operator A, such that μ_t(A) > 0 for all t > 0 and assume that μ_2t(A) / μ_t(A) →1 as t→∞. Let a τ-compact operator S be so that the operator I+S is right invertible, where I is the unit of M. Then, for a τ-measurable operator B, such that A=B(I+S), we have μ_t(A) / μ_t(B) →1 as t→∞. It is an analog of the M.G. Krein theorem (for M=B(H) and τ=tr, theorem 11.4, ch. V [Gohberg I.C., Krein M.G. Introduction to the theory of linear nonselfadjoint operators. In: Translations of Mathematical Monographs. Vol. 18. Providence, R.I., Amer. Math. Soc., 1969. 378 p.] for τ-measurable operators.

Keywords: Hilbert space, von Neumann algebra, normal trace, τ-measurable operator, distribution function, rearrangement, τ-compact operator

 Acknowledgements. This work was supported by subsidies allocated to Kazan Federal University for the state assignment in the sphere of scientific activities (projects nos. 1.1515.2017/4.6 and 1.9773.2017/8.9).

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For citation: Bikchentaev A.M. On an analog of the M.G. Krein theorem for measurable operators.  Uchenye  Zapiski  Kazanskogo  Universiteta.  Seriya  Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 2, pp. 243–249.

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