Speaker
Description
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 $\ast$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 $S({\mathcal M}, \tau )$ with ${\mathcal P}_1\cup {\mathcal P}_3 \subset {\mathcal P}_2$ and investigate their properties.
If an operator $T\in S({\mathcal M}, \tau ) $ is $p$hyponormal for $0 < p \le 1$, 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 $k=1,2, 3$; if an operator $T$ from ${\mathcal P}_1$ admits the bounded inverse $T^{1}$ then $T^{1}$ lies in ${\mathcal P}_1$. If a bounded operator $T$ lies in $\mathcal{P}_1\cup {\mathcal P}_3$ then $T$ is normaloid. If an $T\in S({\mathcal M}, \tau ) $ is hyponormal and $T^n $ is $\tau$compact operator for some natural number $n$ then $T$ is both normal and $\tau$compact. If an operator $T$ lies in $\mathcal{P}_1$ then $T^2$ belongs to $\mathcal{P}_1$. If $\mathcal{M}=\mathcal{B}(\mathcal{H})$ and $\tau={\mathrm tr}$ is the canonical trace, then the class $ \mathcal{P}_1 $ (resp., $ \mathcal{P}_3 $) coincides with the set of all paranormal (resp., $\ast$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 [1]. We also investigate some properties of the KaltonSukochev uniform majorization in $S({\mathcal M}, \tau )$ [2].
The work performed under the development program of Volga Region Mathematical Center
(agreement no. 0750220211393).
REFERENCES

Bikchentaev A. Paranormal measurable operators affiliated with a semifinite von Neumann algebra. II.
Positivity 24 (2020), no. 5, 14871501. 
Bikchentaev A., Sukochev F. Inequalities for the block projection operators. J.
Funct. Anal. 280 (2021), no. 7, 108851, 18 pp.