# Weak almost periodicity of ${L}_{1}$ contractions and coboundaries of non-singular transformations

Studia Mathematica (2000)

- Volume: 138, Issue: 3, page 225-240
- ISSN: 0039-3223

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topKornfeld, Isaac, and Lin, Michael. "Weak almost periodicity of $L_1$ contractions and coboundaries of non-singular transformations." Studia Mathematica 138.3 (2000): 225-240. <http://eudml.org/doc/216701>.

@article{Kornfeld2000,

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 ⨍ ≡ 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)},

author = {Kornfeld, Isaac, Lin, Michael},

journal = {Studia Mathematica},

keywords = {positive contraction; weakly almost periodic; mean ergodic; multiplicative coboundaries; weakly mixing transformation},

language = {eng},

number = {3},

pages = {225-240},

title = {Weak almost periodicity of $L_1$ contractions and coboundaries of non-singular transformations},

url = {http://eudml.org/doc/216701},

volume = {138},

year = {2000},

}

TY - JOUR

AU - Kornfeld, Isaac

AU - Lin, Michael

TI - Weak almost periodicity of $L_1$ contractions and coboundaries of non-singular transformations

JO - Studia Mathematica

PY - 2000

VL - 138

IS - 3

SP - 225

EP - 240

AB - 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)

LA - eng

KW - positive contraction; weakly almost periodic; mean ergodic; multiplicative coboundaries; weakly mixing transformation

UR - http://eudml.org/doc/216701

ER -

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