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Ergodic results for certain contractions on Orlicz spaces with fixed points. (English) Zbl 0694.47007

Author’s abstract: Let (X,\({\mathcal M},\mu)\) be a \(\sigma\)-finite measure space, \(L_{\phi}\equiv L_{\phi}(X,{\mathcal M},\mu)\) an Orlicz space associated to an N-function \(\phi\) and let T: \(L_{\phi}\to L_{\phi}\) be a linear operator with a fixed point \(h\neq 0\) a.e., such that \[ \int_{x}\phi (| Tf|)d\mu \leq \int_{x}\phi (| f|)d\mu \quad (f\in L_{\phi}) \] and it is either a \(\| \|_ 1\)- contraction in \(L_{\phi}\cap L_ 1\) or a \(\| \|_{\infty}\)- contraction in \(L_{\phi}\cap L_{\infty}\). The main result of this paper is that for a wide class of N-functions \(\phi\), the ergodic maximal operator associated to T is bounded in \(L_{\phi}\). Moreover, for every \(f\in L_{\phi}\) we have the almost everywhere convergence and the norm convergence of certain weighted averages which include the Césàro averages.
Reviewer: D.Maharam-Stone

MSC:

47A35 Ergodic theory of linear operators
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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