Portal de Periódicos da CAPES

Remarks on uniform ergodic theorems

2015; Birkhäuser; Volume: 81; Issue: 1-2 Linguagem: Inglês

10.14232/actasm-012-307-4

2064-8316

Michael Lin, David Shoikhet, Laurain Suciu,

Advanced Topology and Set Theory

Let T be a bounded linear operator on a Banach space $$\mathcal{X}$$ In this paper we study uniform Cesàro ergodicity when T is not necessarily powerbounded, and relate it to the uniform convergence of the Abel averages. When $$\mathcal{X}$$ 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 Cesàro ergodicity. An example shows that this is not true in general. For a C0-semi-group {}t≥0 on $$\mathcal{X}$$ complex satisfying limt→∞∥Ti∥/t=0, we show that uniform ergodicity is equivalent to uniform convergence of (λRλ)n for every (one) λ > 0, where Rλ is the resolvent family of the generator of the semi-group.