Uniform ergodicity and the one-sided ergodic Hilbert transform

2023/10/24 by Guy Cohen, Michael Lin, Cohen, Guy +1
Mathematics · #Spectral Theory in Mathematical Physics #advanced mathematical theories #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.2310.15561

Abstract

Let T be a bounded linear operator on a Banach space X satisfying ‖Tⁿ‖/n → 0. We prove that T is uniformly ergodic if and only if the one-sided ergodic Hilbert transform H_Tx:= lim_n→∞ ∑_k=1ⁿ k^-1T^k x converges for every x ∈ (I-T)X. When T is power-bounded (or more generally (C,α) bounded for some 0< α<1), then T is uniformly ergodic if and only if the domain of H_T equals (I-T)X. We then study rotational uniform ergodicity -- uniform ergodicity of every λT with |λ|=1, and connect it to convergence of the rotated one-sided ergodic Hilbert transform, H_λTx. In the Appendix we prove that positive isometries with finite-dimensional fixed space on infinite-dimensional Banach lattices are never uniformly ergodic. In particular, the Koopman operators of ergodic, even non-invertible, probability preserving transformations on standard spaces are never uniformly ergodic.

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