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SUMMARY:On measurable operators affiliated to semifinite von Neumann algeb
 ras
DTSTART;VALUE=DATE-TIME:20210702T120000Z
DTEND;VALUE=DATE-TIME:20210702T123000Z
DTSTAMP;VALUE=DATE-TIME:20260414T003444Z
UID:indico-contribution-198@indico.eimi.ru
DESCRIPTION:Speakers: Airat Bikchentaev (Kazan Federal University)\nLet  $
 \\mathcal{M}$  be a von Neumann algebra of operators on a Hilbert space ${
 \\mathcal H}$ and $\\tau$ be a faithful normal semifinite trace on ${\\mat
 hcal M}$. Let $t_{\\tau}$ be the measure topology  on the $\\ast$-algebra 
 $S({\\mathcal M}\, \\tau )$  of all $\\tau$-measurable operators. We  defi
 ne 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 pr
 operties.\n\nIf an  operator  $T\\in S({\\mathcal M}\, \\tau ) $ is $p$-hy
 ponormal for $0 < p \\le 1$\, then $T$ lies in ${\\mathcal P}_1$\; if an o
 perator $T$ lies in ${\\mathcal P}_k$\, then $UTU^*$ belongs to ${\\mathca
 l P}_k$ 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 a
 n   $T\\in S({\\mathcal M}\, \\tau ) $ is  hyponormal and $T^n $ is  $\\ta
 u$-compact operator for some natural number $n$ then  $T$ is both normal a
 nd $\\tau$-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 al
 l paranormal (resp.\,  $\\ast$-paranormal) operators on $\\mathcal{H}$. Le
 t $A\, B \\in S({\\mathcal M}\, \\tau )$ 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 ma
 jorization in $S({\\mathcal M}\, \\tau )$ [2].\n\nThe work performed under
  the development program of Volga Region Mathematical Center\n(agreement n
 o. 075-02-2021-1393).\n\nREFERENCES\n\n1. Bikchentaev A. Paranormal measur
 able operators affiliated with a semifinite von Neumann algebra. II.\nPosi
 tivity 24 (2020)\, no. 5\, 1487--1501.\n\n2. Bikchentaev A.\, Sukochev F. 
 Inequalities for the block projection operators. J.\nFunct. Anal. 280 (202
 1)\, no. 7\, 108851\, 18 pp.\n\nhttps://indico.eimi.ru/event/321/contribut
 ions/198/
LOCATION:
URL:https://indico.eimi.ru/event/321/contributions/198/
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