Kazan (Volga region) Federal University, KFU
KAZAN
FEDERAL UNIVERSITY
 
ON OPERATORS ALL OF WHICH POWERS HAVE THE SAME TRACE
Form of presentationArticles in international journals and collections
Year of publication2019
Языканглийский
  • 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

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