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# On convexity and compactness of operator intervals'' on Hilbert space

 dc.contributor Казанский федеральный университет dc.contributor.author Bikchentaev Airat Midkhatovich dc.date.accessioned 2018-02-06T13:33:28Z dc.date.available 2018-02-06T13:33:28Z dc.date.issued 2018 dc.identifier.citation Bikchentaev A.M., On convexity and compactness of operator "intervals'' on Hilbert space / A.M. Bikchentaev // Internat. sci. confer. "Infinite-dimensional analysis and control theory" dedicated to the centenary of the outstanding Russian mathematician S.V. Fomin (Moscow, January 29 - February 01, 2018). - M., 2018. - P. 4. dc.identifier.uri http://dspace.kpfu.ru/xmlui/handle/net/118018 dc.description.abstract We consider a von Neumann algebra $M$ acting on a Hilbert space $H$. For a positive operator $X$ in $M$ we define the operator intervals'' $I_X=\{Y=Y^*\in M: \; -X \leq Y \leq X \}$ and $L_X=\{Y \in M: \; |Y| \leq X \}$, where $|Y|=\sqrt{Y^*Y}$. The properties of this operator intervals'' are investigated. We prove that a von Neumann algebra $M$ is Abelian if and only if $L_X$ is convex for all $X$ in $M$. We then show for $M=B(H)$, the algebra of all linear bounded operators on $H$, that the operator interval'' $I_X$ is compact if and only if an operator $X$ is compact. dc.language.iso en dc.relation.ispartofseries Internat. sci. confer. andquot;Infinite-dimensional analysis and control theoryandquot; dedicated to the centenary of the outstanding Russian mathematician S.V. Fomin (Moscow, January 29 - February 01, 2018) dc.rights открытый доступ dc.subject Hilbert space dc.subject von Neumann algebra dc.subject operator order dc.subject convexity dc.subject compactness dc.subject.other Математика dc.title On convexity and compactness of operator intervals'' on Hilbert space dc.type Thesis dc.contributor.org Институт вычислительной математики и информационных технологий dc.description.pages dc.pub-id 173945
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