KFU. Card publication. On $\tau$-essentially invertibility of $\tau$-measurable operators

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FEDERAL UNIVERSITY

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ON $\TAU$-ESSENTIALLY INVERTIBILITY OF $\TAU$-MEASURABLE OPERATORS

Form of presentation

Articles in international journals and collections

Year of publication

2019

Язык

армянский

Authors, employees of KFU

Bikchentaev Ayrat Midkhatovich, author

Bibliographic description in the original language

Airat M. Bikchentaev, On $\tau$-essentially invertibility of $\tau$-measurable operators // Internat. J. Theor. Phys. 2019. V. 58. No 12. https://doi.org/10.1007/s10773-019-04111-w

Annotation

Let ${\mathcal M}$ be a von Neumann algebra of operators on a Hilbert space and $\tau$ be a faithful normal semifinite trace on $\mathcal{M}$. Let $I$ be the unit of the algebra ${\mathcal M}$. A $\tau$-measurable operator $A$ is said to be {\it $\tau$-essentially right (or left) invertible} if there exists a $\tau$-measurable operator $B$ such that the operator $I-AB$ (or $I-BA$) is $\tau$-compact. A necessary and sufficient condition for an operator $A $ to be $\tau$-essentially left invertible is that $A^*A$ (or, equivalently, $\sqrt{A^*A}$) is $\tau$-essentially invertible. We present a sufficient condition that a $\tau$-measurable operator $A $ not be $\tau$-essentially left invertible. For $\tau$-measurable operators $A$ and $P=P^2$ the following conditions are equivalent: 1. $A$ is $\tau$-essential right inverse for $P$; 2. $A$ is $\tau$-essential left inverse for $P$; 3. $I-A, I-P $ are $\tau$-compact; 4. $PA$ is $\tau$-essential left inverse for $P$. For $\tau$-measurable operators $A=A^3$, $ B=B^3$ the following conditions are equivalent: 1. $B$ is $\tau$-essential right inverse for $A$; 2. $B$ is $\tau$-essential left inverse for~$A$. Pairs of faithful normal semifinite traces on $\mathcal{M}$ are considered.

Keywords

Hilbert space, von Neumann algebra, normal weight, semifinite trace, measure topology, $\tau$-measurable operator, $\tau$-compact operator, rearrangement, $\tau$-essentially invertible operator, idempotent.

The name of the journal

INT J THEOR PHYS

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

https://repository.kpfu.ru/eng/?p_id=202091&p_lang=2

Resource files

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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	Airat M. Bikchentaev, On $\tau$-essentially invertibility of $\tau$-measurable operators // Internat. J. Theor. Phys. 2019. V. 58. No 12. https://doi.org/10.1007/s10773-019-04111-w	ru_RU
dc.identifier.uri	https://repository.kpfu.ru/eng/?p_id=202091&p_lang=2	ru_RU
dc.description.abstract	INT J THEOR PHYS	ru_RU
dc.description.abstract	Let ${\mathcal M}$ be a von Neumann algebra of operators on a Hilbert space and $\tau$ be a faithful normal semifinite trace on $\mathcal{M}$. Let $I$ be the unit of the algebra ${\mathcal M}$. A $\tau$-measurable operator $A$ is said to be {\it $\tau$-essentially right (or left) invertible} if there exists a $\tau$-measurable operator $B$ such that the operator $I-AB$ (or $I-BA$) is $\tau$-compact. A necessary and sufficient condition for an operator $A $ to be $\tau$-essentially left invertible is that $A^A$ (or, equivalently, $\sqrt{A^A}$) is $\tau$-essentially invertible. We present a sufficient condition that a $\tau$-measurable operator $A $ not be $\tau$-essentially left invertible. For $\tau$-measurable operators $A$ and $P=P^2$ the following conditions are equivalent: 1. $A$ is $\tau$-essential right inverse for $P$; 2. $A$ is $\tau$-essential left inverse for $P$; 3. $I-A, I-P $ are $\tau$-compact; 4. $PA$ is $\tau$-essential left inverse for $P$. For $\tau$-measurable operators $A=A^3$, $ B=B^3$ the following conditions are equivalent: 1. $B$ is $\tau$-essential right inverse for $A$; 2. $B$ is $\tau$-essential left inverse for~$A$. Pairs of faithful normal semifinite traces on $\mathcal{M}$ are considered.	ru_RU
dc.language.iso	ru	ru_RU
dc.subject	Hilbert space	ru_RU
dc.subject	von Neumann algebra	ru_RU
dc.subject	normal weight	ru_RU
dc.subject	semifinite trace	ru_RU
dc.subject	measure topology	ru_RU
dc.subject	$\tau$-measurable operator	ru_RU
dc.subject	$\tau$-compact operator	ru_RU
dc.subject	rearrangement	ru_RU
dc.subject	$\tau$-essentially invertible operator	ru_RU
dc.subject	idempotent.	ru_RU
dc.title	On $\tau$-essentially invertibility of $\tau$-measurable operators	ru_RU
dc.type	Articles in international journals and collections	ru_RU