Mathematical Research Letters

Volume 18 (2011)

Number 2

On rates in mean ergodic theorems

Pages: 201 – 213



Alexander Gomilko (Faculty of Mathematics and Computer Science, Nicolas Copernicus University, ul. Chopina 12/18, 87-100 Torun, Poland)

Markus Haase (Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, 00-956 Warszawa, Poland)

Yuri Tomilov (Delft Institute of Applie)


We create a general framework for the study of rates of decay in mean ergodic theorems. As a result, we unify and generalize results due to Assani, Cohen, Cuny, Derriennic, and Lin dealing with rates in mean ergodic theorems in a number of cases. In particular, we prove that the Ces\`aro means of a power-bounded operator applied to elements from the domain of its abstract one-sided ergodic Hilbert transform decay logarithmically, and this decay is best possible under natural spectral assumptions.

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