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Studia Mathematica

2000 | 138 | 3 | 225-240
Tytuł artykułu

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

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EN
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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)
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
225-240
Opis fizyczny
Daty
wydano
2000
otrzymano
1998-08-31
poprawiono
1999-11-03
Twórcy
autor
autor
Bibliografia
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• [ÇLO] D. Çömez, M. Lin, and J. Olsen, Weighted ergodic theorems for mean ergodic $L_1$ contractions, Trans. Amer. Math. Soc. 350 (1998), 101-117.
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• [HOkOs] T. Hamachi, Y. Oka, and M. Osikawa, A classification of ergodic non-singular transformation groups, Mem. Fac. Sci. Kyushu Univ. Ser. A 28 (1974), 113-133.
• [He-1] H. Helson, Note on additive cocycles, J. London Math. Soc. 31 (1985), 473-477.
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• [Y] K. Yosida, Functional Analysis, 3rd ed., Springer, Berlin, 1971.
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