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

A.M. Bikchentaev

Kazan Federal University, Kazan, 420008 Russia

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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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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Received

October 12, 2017

Bikchentaev Airat Midkhatovich, Doctor of Physical and Mathematical Sciences, Leading Research Fellow of Department of Theory of Functions and Approximations

Kazan Federal University

ul. Kremlevskaya, 18, Kazan, 420008 Russia

E-mail: Airat.Bikchentaev@kpfu.ru

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