KFU. Card publication. On the $\tau$-compactness of products of $\tau$-measurable operators adjoint to semi-finite von Neumann algebras

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ON THE $\TAU$-COMPACTNESS OF PRODUCTS OF $\TAU$-MEASURABLE OPERATORS ADJOINT TO SEMI-FINITE VON NEUMANN ALGEBRAS

Form of presentation

Articles in international journals and collections

Year of publication

2019

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Authors, employees of KFU

Bikchentaev Ayrat Midkhatovich, author

Bibliographic description in the original language

Bikchentaev A.M. On the $\tau$-compactness of products of $\tau$-measurable operators adjoint to semi-finite von Neumann algebras / A.M. Bikchentaev // Journal of Mathematical Sciences. - 2019. - 241 (4). - P. 458-468.

Annotation

Let M be the von Neumann algebra of operators in a Hilbert space H and $\tau$ be an exact normal semi-finite trace on M. We obtain inequalities for permutations of products of $\tau$-measurable operators. We apply these inequalities to obtain new submajorizations (in the sense of Hardy, Littlewood, and P´olya) of products of $\tau$-measurable operators and a sufficient condition of orthogonality of certain nonnegative $\tau$-measurable operators. We state sufficient conditions of the $\tau$-compactness of products of self-adjoint $\tau$-measurable operators and obtain a criterion of the $\tau$-compactness of the product of a nonnegative $\tau$-measurable operator and an arbitrary $\tau$-measurable operator. We present an example that shows that the nonnegativity of one of the factors is substantial. We also state a criterion of the elementary nature of the product of nonnegative operators from M. All results are new for the *-algebra B(H) of all bounded linear operators in H endowed with the canonical trace $\tau$ = tr.

Keywords

Hilbert space, linear operator, von Neumann algebra, normal semi-finite trace, $\tau$-measurable operator, $\tau$-compact operator, elementary operator, nilpotent, permutation, submajorization.

The name of the journal

Journal of Mathematical Sciences (United States)

On-line resource for training course

http://dspace.kpfu.ru/xmlui/bitstream/handle/net/151521/Bikchentaev_2019_Journal_of_Mathematical_Sciences.pdf?sequence=1&isAllowed=y

Please use this ID to quote from or refer to the card

https://repository.kpfu.ru/eng/?p_id=205879&p_lang=2

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Full metadata record

Field DC	Value	Language
dc.contributor.author	Bikchentaev Ayrat Midkhatovich	ru_RU
dc.date.accessioned	2019-01-01T00:00:00Z	ru_RU
dc.date.available	2019-01-01T00:00:00Z	ru_RU
dc.date.issued	2019	ru_RU
dc.identifier.citation	Bikchentaev A.M. On the $\tau$-compactness of products of $\tau$-measurable operators adjoint to semi-finite von Neumann algebras / A.M. Bikchentaev // Journal of Mathematical Sciences. - 2019. - 241 (4). - P. 458-468.	ru_RU
dc.identifier.uri	https://repository.kpfu.ru/eng/?p_id=205879&p_lang=2	ru_RU
dc.description.abstract	Journal of Mathematical Sciences (United States)	ru_RU
dc.description.abstract	Let M be the von Neumann algebra of operators in a Hilbert space H and $\tau$ be an exact normal semi-finite trace on M. We obtain inequalities for permutations of products of $\tau$-measurable operators. We apply these inequalities to obtain new submajorizations (in the sense of Hardy, Littlewood, and P´olya) of products of $\tau$-measurable operators and a sufficient condition of orthogonality of certain nonnegative $\tau$-measurable operators. We state sufficient conditions of the $\tau$-compactness of products of self-adjoint $\tau$-measurable operators and obtain a criterion of the $\tau$-compactness of the product of a nonnegative $\tau$-measurable operator and an arbitrary $\tau$-measurable operator. We present an example that shows that the nonnegativity of one of the factors is substantial. We also state a criterion of the elementary nature of the product of nonnegative operators from M. All results are new for the *-algebra B(H) of all bounded linear operators in H endowed with the canonical trace $\tau$ = tr.	ru_RU
dc.language.iso	ru	ru_RU
dc.subject	Hilbert space	ru_RU
dc.subject	linear operator	ru_RU
dc.subject	von Neumann algebra	ru_RU
dc.subject	normal semi-finite trace	ru_RU
dc.subject	$\tau$-measurable operator	ru_RU
dc.subject	$\tau$-compact operator	ru_RU
dc.subject	elementary operator	ru_RU
dc.subject	nilpotent	ru_RU
dc.subject	permutation	ru_RU
dc.subject	submajorization.	ru_RU
dc.title	On the $\tau$-compactness of products of $\tau$-measurable operators adjoint to semi-finite von Neumann algebras	ru_RU
dc.type	Articles in international journals and collections	ru_RU