## Abstract

It is well known that a weakly almost periodic operator T in a Banach space is mean ergodic, and in the complex case, also λT is mean ergodic for every |λ| = 1. We prove that a positive contraction on L_{1} is weakly almost periodic if (and only if) it is mean ergodic. An example shows that without positivity the result is false. In order to construct a contraction T on a complex L_{1} such that λT is mean ergodic whenever |λ| = 1, but T is not weakly almost periodic, we prove the following: Let τ be an invertible weakly mixing non-singular transformation of a separable atomless probability space. Then there exists a complex function φ ∈ L_{∞} with |φ(x)| = 1 a.e. such that for every λ ∈ ℂ with |λ| = 1 the function f ≡ 0 is the only solution of the equation f(τx) = λφ(x)f(x). Moreover, the set of such functions φ is residual in the set of all complex unimodular measurable functions (with the L_{1} topology).

Original language | English |
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Pages (from-to) | 225-240 |

Number of pages | 16 |

Journal | Studia Mathematica |

Volume | 138 |

Issue number | 3 |

State | Published - 1 Dec 2000 |

## ASJC Scopus subject areas

- Mathematics (all)

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