RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Subscription
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Izv. RAN. Ser. Mat.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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

This article is cited in 6 scientific papers (total in 6 papers)

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

Full text: PDF file (549 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 517.98
MSC: Primary 47A35; Secondary 47A84
Received: 08.02.2010
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
\Bibitem{Rom11}
\by A.~V.~Romanov
\paper Weak${}^*$ convergence of operator means
\jour Izv. RAN. Ser. Mat.
\yr 2011
\vol 75
\issue 6
\pages 79--98
\mathnet{http://mi.mathnet.ru/izv4406}
\crossref{https://doi.org/10.4213/im4406}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2918894}
\zmath{https://zbmath.org/?q=an:1248.47013}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2011IzMat..75.1165R}
\elib{http://elibrary.ru/item.asp?id=20358819}
\transl
\jour Izv. Math.
\yr 2011
\vol 75
\issue 6
\pages 1165--1183
\crossref{https://doi.org/10.1070/IM2011v075n06ABEH002568}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000298497200004}
\elib{http://elibrary.ru/item.asp?id=18031630}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84455199899}


Linking options:
  • http://mi.mathnet.ru/eng/izv4406
  • https://doi.org/10.4213/im4406
  • http://mi.mathnet.ru/eng/izv/v75/i6/p79

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. A. V. Romanov, “Ergodic properties of discrete dynamical systems and enveloping semigroups”, Ergod. Th. Dynam. Sys., 36:1 (2016), 198–214  crossref  mathscinet  zmath  isi  scopus
    3. Aleman A., Suciu L., “On Ergodic Operator Means in Banach Spaces”, Integr. Equ. Oper. Theory, 85:2 (2016), 259–287  crossref  mathscinet  zmath  isi  elib  scopus
    4. Suciu L., “Ergodic behaviors of the regular operator means”, Banach J. Math. Anal., 11:2 (2017), 239–265  crossref  mathscinet  zmath  isi
    5. Kreidler H., “Compact Operator Semigroups Applied to Dynamical Systems”, Semigr. Forum, 97:3 (2018), 523–547  crossref  mathscinet  zmath  isi
    6. A. V. Romanov, “Ergodic Properties of Tame Dynamical Systems”, Math. Notes, 106:2 (2019), 286–295  mathnet  crossref  crossref  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
    Number of views:
    This page:364
    Full text:86
    References:35
    First page:14

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020