КФУ. Карточка публикации. On operators all of which powers have the same trace

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ON OPERATORS ALL OF WHICH POWERS HAVE THE SAME TRACE

Форма представления

Статьи в зарубежных журналах и сборниках

Год публикации

2019

Язык

английский

Авторы, сотрудники КФУ

Бикчентаев Айрат Мидхатович, автор

Иваньшин Петр Николаевич, автор

Библиографическое описание на языке оригинала

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

Аннотация

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 $\mathbb{M}_2(\mathbb{C})$.

Ключевые слова

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

Название журнала

INT J THEOR PHYS

URL

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

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https://repository.kpfu.ru/?p_id=197010

Полная запись метаданных

Поле DC	Значение	Язык
dc.contributor.author	Бикчентаев Айрат Мидхатович	ru_RU
dc.contributor.author	Иваньшин Петр Николаевич	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/?p_id=197010	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	Статьи в зарубежных журналах и сборниках	ru_RU