# On a vector-valued local ergodic theorem in ${L}_{\infty}$

Studia Mathematica (1999)

- Volume: 132, Issue: 3, page 285-298
- ISSN: 0039-3223

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topSato, Ryotaro. "On a vector-valued local ergodic theorem in $L_∞$." Studia Mathematica 132.3 (1999): 285-298. <http://eudml.org/doc/216600>.

@article{Sato1999,

abstract = {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*.},

author = {Sato, Ryotaro},

journal = {Studia Mathematica},

keywords = {vector-valued local ergodic theorem; reflexive Banach space; d-dimensional semigroup of linear contractions; contraction majorant; semigroup of linear contractions; local ergodic theorem},

language = {eng},

number = {3},

pages = {285-298},

title = {On a vector-valued local ergodic theorem in $L_∞$},

url = {http://eudml.org/doc/216600},

volume = {132},

year = {1999},

}

TY - JOUR

AU - Sato, Ryotaro

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

JO - Studia Mathematica

PY - 1999

VL - 132

IS - 3

SP - 285

EP - 298

AB - 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*.

LA - eng

KW - vector-valued local ergodic theorem; reflexive Banach space; d-dimensional semigroup of linear contractions; contraction majorant; semigroup of linear contractions; local ergodic theorem

UR - http://eudml.org/doc/216600

ER -

## References

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