## 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 A_{α}, α ∈ (0, 1). This is equivalent to uniform Cesàro ergodicity of T when ∥T^{n}∥/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 C_{0}-semi-group {Tt}_{t≥0} on X complex satisfying limt_{t→∞}/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 language | English |
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Pages (from-to) | 251-283 |

Number of pages | 33 |

Journal | Acta Scientiarum Mathematicarum |

Volume | 81 |

Issue number | 1-2 |

DOIs | |

State | Published - 1 Jan 2015 |

## Keywords

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

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics