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A remark on the ergodic Hilbert transform. (English) Zbl 0641.47008

Let T be an invertible power bounded Lamperti operator of \(L_ p(X)\), \(1<p<\infty\). The powers of such a T can be represented in the form \((T^ kf)(x)=h_ k(x)\Phi^ k(x)\), where \(f\in L_ p(X)\), \(\Phi\) is a measure-preserving transformation on X, and the \(h_ k\) are appropriate measurable functions on X. The author defines (\(\tau\) f)(x) as \(| h_ 1(x)| \Phi (x)\), and shows that \(\tau\) is an invertible operator on \(L_ p\) such that \((\tau^ kf)(x)=| h_ k(x)| f(\Phi^ kx)\). He then proves that if \(\sup_{n\geq 0}\| 1/(2n+1)\sum^{n}_{k=-n}\tau^ k\|_ p<\infty\), then the ergodic Hilbert transform Hf exists a.e. on X.
Reviewer: D.Maharam-Stone

MSC:

47A35 Ergodic theory of linear operators
44A15 Special integral transforms (Legendre, Hilbert, etc.)
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