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.