KFU. Card publication. On operators all of which powers have the same trace

KAZAN
FEDERAL UNIVERSITY

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ON OPERATORS ALL OF WHICH POWERS HAVE THE SAME TRACE

Form of presentation

Articles in international journals and collections

Year of publication

2019

Язык

английский

Authors, employees of KFU

Bikchentaev Ayrat Midkhatovich, author

Ivanshin Petr Nikolaevich, author

Bibliographic description in the original language

Airat M. Bikchentaev, Pyotr N. Ivanshin, On operators all of which powers have the same trace, // Internat. J. Theor. Physics, https://doi.org/10.1007/s10773-019-04059-x

Annotation

We introduce the class $K_{\mathcal{A}, \phi}=\{a \in \mathcal{A}: \phi(A^k)=\phi(A)$ for all $k \in \mathbb{N}\}$ for a linear functional $\phi$ on an algebra $\mathcal{A}$ and consider the properties of this class. Also we prove the ``0--1 number lemma'': if a set $\{z_k\}_{k=1}^{n} \subset \mathbb{C}$ is such that $z_1+\ldots+z_n=z_1^2+\ldots+z_n^2=\cdots=z_1^{n+1}+\ldots+z_n^{n+1}$, then $z_k \in \{0, 1\}$, for all $k=1, 2, \ldots, n$. This lemma helps us to show that $\{\phi(A): A \in K_{\mathcal{A}, \phi}\}=\{0, 1, \ldots, n\}$ and $\det (A) \in \{0, 1\}$ for $\mathcal{A}=\mathbb{M}_n(\mathbb{C})$ and $\phi =\mathrm{tr}$, the canonical trace. We have $A=P+Z$ where $P$ is a projection and $Z$ is a nilpotent for any $A \in K_{\mathcal{A}, \phi}$. Assume that for a trace class operator $A$ there exists a constant $C \in \mathbb{C}$ such that $\mathrm{tr}(A^k)=C$ for all $k \in \mathbb{N}$. Then $C \in \mathbb{N}\bigcup \{0\}$ and the spectrum $\sigma(A)$ is a subset of $\{0, 1\}$. Finally we give the description of all the elements of the class $ K_{\mathcal{A}, \phi}$ for $\mathbb{M}_2(\mathbb{C})$.

Keywords

Hilbert space, normed algebra, idempotent, $C^*$-algebra, $W^*$-algebra, linear functional, state, tracial functional, trace class operator, Vandermonde matrix, spectrum, determinant.

The name of the journal

INT J THEOR PHYS

URL

https://doi.org/10.1007/s10773-019-04059-x

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

https://repository.kpfu.ru/eng/?p_id=197010&p_lang=2

Full metadata record

Field DC	Value	Language
dc.contributor.author	Bikchentaev Ayrat Midkhatovich	ru_RU
dc.contributor.author	Ivanshin Petr Nikolaevich	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, Pyotr N. Ivanshin, On operators all of which powers have the same trace, // Internat. J. Theor. Physics, https://doi.org/10.1007/s10773-019-04059-x	ru_RU
dc.identifier.uri	https://repository.kpfu.ru/eng/?p_id=197010&p_lang=2	ru_RU
dc.description.abstract	INT J THEOR PHYS	ru_RU
dc.description.abstract	We introduce the class $K_{\mathcal{A}, \phi}=\{a \in \mathcal{A}: \phi(A^k)=\phi(A)$ for all $k \in \mathbb{N}\}$ for a linear functional $\phi$ on an algebra $\mathcal{A}$ and consider the properties of this class. Also we prove the ``0--1 number lemma'': if a set $\{z_k\}_{k=1}^{n} \subset \mathbb{C}$ is such that $z_1+\ldots+z_n=z_1^2+\ldots+z_n^2=\cdots=z_1^{n+1}+\ldots+z_n^{n+1}$, then $z_k \in \{0, 1\}$, for all $k=1, 2, \ldots, n$. This lemma helps us to show that $\{\phi(A): A \in K_{\mathcal{A}, \phi}\}=\{0, 1, \ldots, n\}$ and $\det (A) \in \{0, 1\}$ for $\mathcal{A}=\mathbb{M}_n(\mathbb{C})$ and $\phi =\mathrm{tr}$, the canonical trace. We have $A=P+Z$ where $P$ is a projection and $Z$ is a nilpotent for any $A \in K_{\mathcal{A}, \phi}$. Assume that for a trace class operator $A$ there exists a constant $C \in \mathbb{C}$ such that $\mathrm{tr}(A^k)=C$ for all $k \in \mathbb{N}$. Then $C \in \mathbb{N}\bigcup \{0\}$ and the spectrum $\sigma(A)$ is a subset of $\{0, 1\}$. Finally we give the description of all the elements of the class $ K_{\mathcal{A}, \phi}$ for $\mathbb{M}_2(\mathbb{C})$.	ru_RU
dc.language.iso	ru	ru_RU
dc.subject	Hilbert space	ru_RU
dc.subject	normed algebra	ru_RU
dc.subject	idempotent	ru_RU
dc.subject	$C^*$-algebra	ru_RU
dc.subject	$W^*$-algebra	ru_RU
dc.subject	linear functional	ru_RU
dc.subject	state	ru_RU
dc.subject	tracial functional	ru_RU
dc.subject	trace class operator	ru_RU
dc.subject	Vandermonde matrix	ru_RU
dc.subject	spectrum	ru_RU
dc.subject	determinant.	ru_RU
dc.title	On operators all of which powers have the same trace	ru_RU
dc.type	Articles in international journals and collections	ru_RU