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Вещественный, комплексный и и функциональный анализ
Ergodic theorems in Banach ideals of compact operators
A. N. Azizov, V. I. Chilin National University of Uzbekistan, 4, Universitet str., Tashkent, 100174, Uzbekistan
Аннотация:
Let $\mathcal H$ be an infinite-dimensional Hilbert space, and let $\mathcal B(\mathcal H)$ ($\mathcal K(\mathcal H)$) be the $C^\star$–algebra of all bounded (compact) linear operators in $\mathcal H$. Let $(E,\|\cdot\|_E)$ be a fully symmetric sequence space. If $\{s_n(x)\}_{n=1}^\infty$ are the singular values of $x\in\mathcal K(\mathcal H)$, let $\mathcal C_E=\{x\in\mathcal K(\mathcal H): \{s_n(x)\}\in E\}$ with $\|x\|_{\mathcal C_E}=\|\{s_n(x)\}\|_E$, $x\in\mathcal C_E$, be the Banach ideal of compact operators generated by $E$. We show that the averages $A_n(T)(x)=\frac1{n+1}\sum\limits_{k = 0}^n T^k(x) $ converge uniformly in $\mathcal C_E$ for any Dunford-Schwartz operator $T$ and $x\in\mathcal C_E$. Besides, if $0\leq x\in\mathcal B(\mathcal H)\setminus\mathcal K(\mathcal H)$, there exists a Dunford-Schwartz operator $T$ such that the sequence $\{A_n(T)(x)\}$ does not converge uniformly. We also show that the averages $A_n(T)$ converge strongly in $(\mathcal C_E, \|\cdot\|_{\mathcal C_E})$ if and only if $E$ is separable and $E \neq l^1$ as sets.
Ключевые слова:
symmetric sequence space, Banach ideal of compact operators, Dunford-Schwartz operator, individual ergodic theorem, mean ergodic theorem.
Поступила 26 февраля 2021 г., опубликована 21 мая 2021 г.
Образец цитирования:
A. N. Azizov, V. I. Chilin, “Ergodic theorems in Banach ideals of compact operators”, Сиб. электрон. матем. изв., 18:1 (2021), 534–547
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/semr1379 https://www.mathnet.ru/rus/semr/v18/i1/p534
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Страница аннотации: | 150 | PDF полного текста: | 67 | Список литературы: | 16 |
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