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Функциональный анализ и его приложения
14 февраля 2019 г. 10:30–11:50, г. Ташкент, Национальный университет Узбекистана, Математический факультет, аудитория А-304, ул. Университетская, 4
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Ergodic theorems in Banach ideals of compact operators
A. N. Azizova, V. I. Chilina, S. N. Litvinovb a National University of Uzbekistan named after M. Ulugbek, Tashkent
b Pennsylvania State University, Department of Mathematics
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Эта страница: | 177 |
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Аннотация:
Let $\mathcal H$ be a complex infinite-dimensional Hilbert space, and let $\mathcal B (\mathcal H)$ $(\mathcal K (\mathcal H))$ be the $C^\ast$-algebra of bounded (respectively, 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 $\mathcal C_E=\{x\in\mathcal K(\mathcal H): \{s_n(x)\}\subset E\}$ 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) = \frac{1}{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 $x\in\mathcal B(\mathcal H)\setminus\mathcal K(\mathcal H)$, there exists a Hermitian 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.
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