Weak almost periodicity of $L_1$ contractions and coboundaries of non-singular transformations
Studia Mathematica, Tome 138 (2000) no. 3, pp. 225-240
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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 ⨍ ≡ 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 $L_1$ topology)
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author = {Isaac Kornfeld and },
title = {Weak almost periodicity of $L_1$ contractions and coboundaries of non-singular transformations},
journal = {Studia Mathematica},
pages = {225--240},
publisher = {mathdoc},
volume = {138},
number = {3},
year = {2000},
doi = {10.4064/sm-138-3-225-240},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/sm-138-3-225-240/}
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Isaac Kornfeld; . Weak almost periodicity of $L_1$ contractions and coboundaries of non-singular transformations. Studia Mathematica, Tome 138 (2000) no. 3, pp. 225-240. doi: 10.4064/sm-138-3-225-240
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