Hyponormal mesurable operators affiliated to a semifinite von Neumann algebra

Abstract

Let τ be a faithful normal semifinite trace on a von Neumann algebra M. We study the cases when a hyponormal τ-measurable operator (or a estriction of it) is normal. We obtain a criterion for the hyponormality of a -measurable operator in terms of its singular value function. The set of all τ-measurable hyponormal operators is closed in the topology of τ -local convergence in measure. This assertion is a generalization of Problem 226 from the book "Halmos P.R., A Hilbert Space Problem Book, Second edition, Springer, New York (1982)" to the setting of unbounded operators. The set of all τ-measurable cohyponormal operators is closed in the topology of τ -local convergence in measure if and only if the von Neumann algebra M is finite.