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

• Autorzy: Isaac Kornfeld , Michael Lin
• Języki publikacji: EN

Abstrakty

EN

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)

Kategorie tematyczne

• 47A35: Ergodic theory
• 28D05: Measure-preserving transformations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2000
• Tom: 138
• Numer: 3
• Strony: 225-240

Daty

wydano

2000

otrzymano

1998-08-31

poprawiono

1999-11-03

Twórcy

autor

Isaac Kornfeld

• North Dakota State University, Fargo, ND 58105, U.S.A.

autor

Michael Lin

• Ben-Gurion University of the Negev, Beer-Sheva, Israel

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