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.

令\（{{\ mathcal {M}}} \）是希尔伯特空间\（{\ mathcal {H}} \）上算子的冯·诺依曼代数，而\（\ tau \）是Hilbert空间上的忠实正态半迹\（\ mathcal {M} \）。令\（t_ \ tau \）为所有\（\ tau \）可测算子的\（* \）-代数\（S（\ mathcal {M}，\ tau）\）上的度量拓扑。我们定义了三个\（t_ \ tau \）封闭类\（{{\ mathcal {P}}} _ 1 \），\（{{\ mathcal {P}}} _ 2 \）和\（{{\ math { P}}} _ 3 \）的\（\ tau蛋白\） -measurable运营商和研究它们的特性。班级\（{{\ mathcal {P}}} _ 2 \）包含\（{{\ mathcal {P}}} _ 1_1 \ cup {{\ mathcal {P}}} _ 3 \）。如果\（\ tau \）可测算子T为伪正态，则T位于\（{{\ mathcal {P}}} _ 1 \ cap {{\ mathcal {P}}} _ 3 \）中；如果操作者Ť在于\（{{\ mathcal {P}}} _ 3 \） ，然后\（UTU ^ * \）属于\（{{\ mathcal {P}}} _ 3 \）对于所有等距ù从\（{{\ mathcal {M}}} \\）。如果有界算子T位于\（\ mathcal {P} _1 \ cup {{\ mathcal {P}}} _ 3 \）中，则T为法线。如果是操作员\（T \ in S（\ mathcal {M}，\ tau）\）是p-伪正弦形，其中\（0 <p \ le 1 \）然后\（T \ in \ mathcal {P} _1 \）。如果\（\ mathcal {M} = \ mathcal {B}（\ mathcal {H}）\）和\（\ tau = \ text {tr} \）是规范跟踪，则类\（\ mathcal {P } _1 \）（resp。，\（{{\ mathcal {P}}} _ 3 \））与\（\ mathcal {H上的所有超自然（resp。，\（* \）- paranormal）运算符的集合} \）。令\（A，B \ in S（{{\ mathcal {M}}}，\ tau）\）和A为\-（0 <p \ le 1 \）的p伪正态。如果AB是\（\ tau \）-紧凑，然后\（A ^ * B \）是\（\ tau \）-紧凑。