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

Sato, Ryotaro

On a vector-valued local ergodic theorem in

L_{\infty}

Sato, Ryotaro

Studia Mathematica, Tome 133 (1999), p. 285-298 / Harvested from The Polish Digital Mathematics Library

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Résumé

Let $T = T (u) : u \in ℝ_{d}^{+}$ 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 $L_{1} ((Ω, Σ, μ); X) * = L_{\infty} ((Ω, Σ, μ); X *)$ , the adjoint semigroup $T * = T * (u) : u \in ℝ_{d}^{+}$ becomes a weak*-continuous semigroup of linear contractions acting on $L_{\infty} ((Ω, Σ, μ); X *)$ . In this paper the local ergodic theorem is studied for the adjoint semigroup T*. Assuming that each T(u), $u \in ℝ_{d}^{+}$ , 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 $‖ T (u) f (ω) ‖ \leq P (u) ‖ f (\cdot) ‖ (ω)$ almost everywhere on Ω for every $⨍ \in L_{1} ((Ω, Σ, μ); X)$ , we prove that the local ergodic theorem holds for T*.

Publié le : 1999-01-01

Zbl 0933.47004

EUDML-ID : urn:eudml:doc:216600

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Sato, Ryotaro. On a vector-valued local ergodic theorem in $L_∞$
            . Studia Mathematica, Tome 133 (1999) pp. 285-298. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv132i3p285bwm/

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