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
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