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## Studia Mathematica

1999 | 132 | 3 | 285-298
Tytuł artykułu

### On a vector-valued local ergodic theorem in $L_∞$

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EN
Abstrakty
EN
Let $T = {T(u): u ∈ ℝ_d^{+}}$ be a strongly continuous d-dimensional semigroup of linear contractions on $L_1((Ω,Σ,μ);X)$, where (Ω,Σ,μ) is a σ-finite measure space and X is a reflexive Banach space. Since $L_1((Ω,Σ,μ);X)* = L_∞((Ω,Σ,μ);X*)$, the adjoint semigroup $T* = {T*(u): u ∈ ℝ_d^{+}}$ becomes a weak*-continuous semigroup of linear contractions acting on $L_∞((Ω,Σ,μ);X*)$. In this paper the local ergodic theorem is studied for the adjoint semigroup T*. Assuming that each T(u), $u ∈ ℝ_d^{+}$, has a contraction majorant P(u) defined on $L_1((Ω,Σ,μ);ℝ)$, that is, P(u) is a positive linear contraction on $L_1((Ω,Σ,μ);ℝ)$ such that $‖T(u)f(ω)‖ ≤ P(u)‖f(·)‖(ω)$ almost everywhere on Ω for every $⨍ ∈ L_1((Ω,Σ,μ);X)$, we prove that the local ergodic theorem holds for T*.
Słowa kluczowe
EN
Kategorie tematyczne
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Rocznik
Tom
Numer
Strony
285-298
Opis fizyczny
Daty
wydano
1999
otrzymano
1997-11-21
Twórcy
autor
Bibliografia
• [1] M. A. Akcoglu and A. del Junco, Differentiation of n-dimensional additive processes, Canad. J. Math. 33 (1981), 749-768.
• [2] M. A. Akcoglu and U. Krengel, A differentiation theorem for additive processes, Math. Z. 163 (1978), 199-210.
• [3] R. V. Chacon and U. Krengel, Linear modulus of a linear operator, Proc. Amer. Math. Soc. 15 (1964), 553-559.
• [4] J. Diestel and J. J. Uhl, Jr., Vector Measures, Amer. Math. Soc., Providence, 1977.
• [5] N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory, Interscience, New York, 1958.
• [6] R. Emilion, Semi-groups in $L_∞$ and local ergodic theorem, Canad. Math. Bull. 29 (1986), 146-153.
• [7] U. Krengel, Ergodic Theorems, de Gruyter, Berlin, 1985.
• [8] W. Rudin, Functional Analysis, McGraw-Hill, New York, 1973.
• [9] R. Sato, Vector valued differentiation theorems for multiparameter additive processes in $L_p$ spaces, Positivity 2 (1998), 1-18.
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