On Normal $\tau$-Measurable Operators Affiliated with Semifinite Von Neumann Algebras

Let $\tau$ be a faithful normal semifinite trace on the von Neumann algebra $\mathcal{M}$, $1 \ge q >0$. The following generalizations of problems 163 and 139 from the book [1] to $\tau$-measurable operators are obtained; it is established that: 1) each $\tau$-compact $q$-hyponormal operator is normal; 2) if a $\tau$-measurable operator $A$ is normal and, for some natural number $n$, the operator $A^n$ is $\tau$-compact, then the operator $A$ is also $\tau$-compact. It is proved that if a $\tau$-measurable operator $A$ is hyponormal and the operator $A^2$ is $\tau$-compact, then the operator $A$ is also $\tau$-compact. A new property of a nonincreasing rearrangement of the product of hyponormal and cohyponormal $\tau$-measurable operators is established. For normal $\tau$-measurable operators $A$ and $B$, it is shown that the nonincreasing rearrangements of the operators $AB$ and $BA$ coincide. Applications of the results obtained to $F$-normed symmetric spaces on $(\mathcal{M},\tau)$ are considered.