Hyponormal Measurable Operators Affiliated to a Semifinite von Neumann Algebra

Let $ \tau $ be a faithful normal semifinite trace on a von Neumann algebra $ \mathcal{M} $ . We study the cases when a hyponormal $ \tau $ -measurable operator (or a restriction of it) is normal. We obtain a criterion for the hyponormality of a $ \tau $ -measurable operator in terms of its singular value function. The set of all $ \tau $ -measurable hyponormal operators is closed in the topology of $ \tau $ -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 $ \tau $ -measurable cohyponormal operators is closed in the topology of $ \tau $ -local convergence in measure if and only if the von Neumann algebra $ \mathcal{M} $ is finite.