Paranormal measurable operators affiliated with a semifinite von Neumann algebra. II

Abstract

Let ${{\mathcal {M}}}$ be a von Neumann algebra of operators on a Hilbert space ${\mathcal {H}}$ and $\tau$ be a faithful normal semifinite trace on $\mathcal {M}$. Let $t_\tau$ be the measure topology on the $*$-algebra $S(\mathcal {M},\tau )$ of all $\tau$-measurable operators. We define three $t_\tau$-closed classes ${{\mathcal {P}}}_1$, ${{\mathcal {P}}}_2$ and ${{\mathcal {P}}}_3$ of $\tau$-measurable operators and investigate their properties. The class ${{\mathcal {P}}}_2$ contains ${{\mathcal {P}}}_1\cup {{\mathcal {P}}}_3$. If a $\tau$-measurable operator T is hyponormal, then T lies in ${{\mathcal {P}}}_1\cap {{\mathcal {P}}}_3$; if an operator T lies in ${{\mathcal {P}}}_3$, then $UTU^*$ belongs to ${{\mathcal {P}}}_3$ for all isometries U from ${{\mathcal {M}}}$. If a bounded operator T lies in $\mathcal {P}_1\cup {{\mathcal {P}}}_3$ then T is normaloid. If an operator $T\in S(\mathcal {M},\tau )$ is p-hyponormal with $0<p\le 1$ then $T\in \mathcal {P}_1$. If $\mathcal {M}=\mathcal {B}(\mathcal {H})$ and $\tau =\text { tr}$ is the canonical trace, then the class $\mathcal {P}_1$ (resp., ${{\mathcal {P}}}_3$) coincides with the set of all paranormal (resp., $*$-paranormal) operators on $\mathcal {H}$. Let $A, B \in S({{\mathcal {M}}},\tau )$ and A be p-hyponormal with $0<p\le 1$. If AB is $\tau$-compact then $A^*B$ is $\tau$-compact.