Remarks on uniform ergodic theorems

Michael Lin, David Shoikhet, Laurian Suciu

Research output: Contribution to journalArticlepeer-review

17 Scopus citations

Abstract

Let T be a bounded linear operator on a Banach space X. In this paper we study uniform Cesaro ergodicity when T is not necessarily powerbounded, and relate it to the uniform convergence of the Abel averages. When X is over the complex field, we show that uniform Abel ergodicity is equivalent to the uniform convergence of the powers of all (one of) the Abel averages Aα, α ∈ (0, 1). This is equivalent to uniform Cesàro ergodicity of T when ∥Tn∥/n → 0. For positive operators on real or complex Banach lattices, uniform Abel ergodicity is equivalent to uniform Cesaro ergodicity. An example shows that this is not true in general. For a C0-semi-group {Tt}t≥0 on X complex satisfying limtt→∞/t = 0, we show that uniform ergodicity is equivalent to uniform convergence of (R)n for every (one) > 0, where R is the resolvent family of the generator of the semi-group.

Original languageEnglish
Pages (from-to)251-283
Number of pages33
JournalActa Scientiarum Mathematicarum
Volume81
Issue number1-2
DOIs
StatePublished - 1 Jan 2015

Keywords

  • Abel convergence
  • Cesàro bounded operators
  • One-point peripheral spectrum
  • Uniform ergodic theorem

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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