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 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 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 with |φ(x)|=1 a.e. such that for every λ ∈ℂ with |λ|=1 the function ⨍ ≡ 0 is the only solution of the equation ⨍(τx)=λφ(x)⨍(x). Moreover, the set of such functions φ is residual in the set of all complex unimodular measurable functions (with the topology)
@article{bwmeta1.element.bwnjournal-article-smv138i3p225bwm,
author = {Isaac Kornfeld and Michael Lin},
title = {Weak almost periodicity of $L\_1$ contractions and coboundaries of non-singular transformations},
journal = {Studia Mathematica},
volume = {141},
year = {2000},
pages = {225-240},
zbl = {0955.28009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv138i3p225bwm}
}
Kornfeld, Isaac; Lin, Michael. Weak almost periodicity of $L_1$ contractions and coboundaries of non-singular transformations. Studia Mathematica, Tome 141 (2000) pp. 225-240. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv138i3p225bwm/
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