Two classes of $\tau$-measurable operators affiliated with a von Neumann algebra

Let $\mathcal{M}$ be a von Neumann algebra of operators on a Hilbert space $\mathcal{H}$, $\tau$ be a faithful normal semifinite trace on $\mathcal{M}$. We define two (closed in the topology of convergence in measure $\tau$) classes $\mathcal{P}_1$ and $\mathcal{P}_2$ of $\tau$-measurable operators and investigate their properties. The class $ \mathcal{P}_1$ is contained in $ \mathcal{P}_2$. If a $\tau$-measurable operator $T$ is hyponormal, then $T$ lies in $ \mathcal{P}_1$; if an operator $T$ lies in $\mathcal{P}_k$, then $UTU^*$ belongs to $ \mathcal{P}_k$ for all isometries $U $ from $\mathcal{M}$ and $k=1,2$; if an operator $T$ from $ \mathcal{P}_1$ has the bounded inverse $T^{-1} $, then $T^{-1}$ lies in $\mathcal{P}_1$. We establish some new inequalities for rearrangements of operators from $ \mathcal{P}_1$. If a $\tau$-measurable operator $T $ is hyponormal and $T^n $ is $\tau$-compact for some natural number $n$, then $T $ is normal and $\tau$-compact. If $\mathcal{M}=\mathcal{B}(\mathcal{H})$ and $\tau=\mathrm{tr}$, then the class $\mathcal{P}_1$ coincides with the set of all paranormal operators on $\mathcal{H}$.