On a vector-valued local ergodic theorem in L ∞

Ryotaro Sato

Studia Mathematica (1999)

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

Abstract

top
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*.

How to cite

top

Sato, 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

top
  1. [1] M. A. Akcoglu and A. del Junco, Differentiation of n-dimensional additive processes, Canad. J. Math. 33 (1981), 749-768. 
  2. [2] M. A. Akcoglu and U. Krengel, A differentiation theorem for additive processes, Math. Z. 163 (1978), 199-210. 
  3. [3] R. V. Chacon and U. Krengel, Linear modulus of a linear operator, Proc. Amer. Math. Soc. 15 (1964), 553-559. Zbl0168.11701
  4. [4] J. Diestel and J. J. Uhl, Jr., Vector Measures, Amer. Math. Soc., Providence, 1977. 
  5. [5] N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory, Interscience, New York, 1958. Zbl0084.10402
  6. [6] R. Emilion, Semi-groups in L ∞ and local ergodic theorem, Canad. Math. Bull. 29 (1986), 146-153. 
  7. [7] U. Krengel, Ergodic Theorems, de Gruyter, Berlin, 1985. 
  8. [8] W. Rudin, Functional Analysis, McGraw-Hill, New York, 1973. 
  9. [9] R. Sato, Vector valued differentiation theorems for multiparameter additive processes in L p spaces, Positivity 2 (1998), 1-18. Zbl0915.47012

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.