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 Kalton--Sukochev uniform majorization in $S({\mathcal M}, \tau )$ [2].
The work performed under the development program of Volga Region Mathematical Center
(agreement no. 075-02-2021-1393).
REFERENCES
-
Bikchentaev A. Paranormal measurable operators affiliated with a semifinite von Neumann algebra. II.
Positivity 24 (2020), no. 5, 1487--1501. -
Bikchentaev A., Sukochev F. Inequalities for the block projection operators. J.
Funct. Anal. 280 (2021), no. 7, 108851, 18 pp.
