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 Izv. RAN. Ser. Mat., 2011, Volume 75, Issue 6, Pages 79–98 (Mi izv4406)

Weak$^*$ convergence of operator means

A. V. Romanov

Moscow State Institute of Electronics and Mathematics (Technical University)

Abstract: For a linear operator $U$ with $\|U^n\| \le \operatorname{const}$ on a Banach space $X$ we discuss conditions for the convergence of ergodic operator nets $T_\alpha$ corresponding to the adjoint operator $U^*$ of $U$ in the $\mathrm{W^*O}$-topology of the space $\operatorname{End} X^*$. The accumulation points of all possible nets of this kind form a compact convex set $L$ in $\operatorname{End} X^*$, which is the kernel of the operator semigroup $G=\overline{\operatorname{co}} \Gamma_0$, where $\Gamma_0=\{U_n^*, n \ge 0\}$. It is proved that all ergodic nets $T_\alpha$ weakly$^*$ converge if and only if the kernel $L$ consists of a single element. In the case of $X=C(\Omega)$ and the shift operator $U$ generated by a continuous transformation $\varphi$ of a metrizable compactum $\Omega$ we trace the relationships among the ergodic properties of $U$, the structure of the operator semigroups $L$, $G$ and $\Gamma=\overline{\Gamma}_0$, and the dynamical characteristics of the semi-cascade $(\varphi,\Omega)$. In particular, if $\operatorname{card}L=1$, then a) for any $\omega \in\Omega$ the closure of the trajectory $\{\varphi^n\omega, n \ge 0\}$ contains precisely one minimal set $m$, and b) the restriction $(\varphi,m)$ is strictly ergodic. Condition a) implies the $\mathrm{W^*O}$-convergence of any ergodic sequence of operators $T_n \in \operatorname{End} X^*$ under the additional assumption that the kernel of the enveloping semigroup $E(\varphi,\Omega)$ contains elements obtained from the ‘basis’ family of transformations $\{\varphi^n, n \ge 0\}$ of the compact set $\Omega$ by using some transfinite sequence of sequential passages to the limit.

Keywords: weak$^*$ ergodic theory, dynamical system, enveloping semigroup, Choquet representation.

DOI: https://doi.org/10.4213/im4406

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English version:
Izvestiya: Mathematics, 2011, 75:6, 1165–1183

Bibliographic databases:

UDC: 517.98
MSC: Primary 47A35; Secondary 47A84
Revised: 22.03.2010

Citation: A. V. Romanov, “Weak$^*$ convergence of operator means”, Izv. RAN. Ser. Mat., 75:6 (2011), 79–98; Izv. Math., 75:6 (2011), 1165–1183

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv4406
• https://doi.org/10.4213/im4406
• http://mi.mathnet.ru/eng/izv/v75/i6/p79

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. V. Romanov, “Ordinary Semicascades and Their Ergodic Properties”, Funct. Anal. Appl., 47:2 (2013), 160–163
2. A. V. Romanov, “Ergodic properties of discrete dynamical systems and enveloping semigroups”, Ergod. Th. Dynam. Sys., 36:1 (2016), 198–214
3. Aleman A., Suciu L., “On Ergodic Operator Means in Banach Spaces”, Integr. Equ. Oper. Theory, 85:2 (2016), 259–287
4. Suciu L., “Ergodic behaviors of the regular operator means”, Banach J. Math. Anal., 11:2 (2017), 239–265
5. Kreidler H., “Compact Operator Semigroups Applied to Dynamical Systems”, Semigr. Forum, 97:3 (2018), 523–547
6. A. V. Romanov, “Ergodic Properties of Tame Dynamical Systems”, Math. Notes, 106:2 (2019), 286–295
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