# Vector-valued ergodic theorems for multiparameter additive processes

Colloquium Mathematicae (1999)

- Volume: 79, Issue: 2, page 193-202
- ISSN: 0010-1354

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topSato, Ryotaro. "Vector-valued ergodic theorems for multiparameter additive processes." Colloquium Mathematicae 79.2 (1999): 193-202. <http://eudml.org/doc/210634>.

@article{Sato1999,

abstract = {Let X be a reflexive Banach space and (Ω,Σ,μ) be a σ-finite measure space. Let d ≥ 1 be an integer and T=T(u):u=($u_\{1\}$, ... ,$u_\{d\})$, $u_\{i\}$ ≥ 0, 1 ≤ i ≤ d be a strongly measurable d-parameter semigroup of linear contractions on $L_\{1\}$((Ω,Σ,μ);X). We assume that to each T(u) there corresponds a positive linear contraction P(u) defined on $L_\{1\}$((Ω,Σ,μ);ℝ) with the property that ∥ T(u)f(ω)∥ ≤ P(u)∥f(·)∥(ω) almost everywhere on Ω for all f ∈ $L_\{1\}$((Ω,Σ,μ);X). We then prove stochastic and pointwise ergodic theorems for a d-parameter bounded additive process F in $L_\{1\}$((Ω,Σ,μ);X) with respect to the semigroup T.},

author = {Sato, Ryotaro},

journal = {Colloquium Mathematicae},

keywords = {multiparameter dynamical system; ergodic theorem; multiparameter additive processes},

language = {eng},

number = {2},

pages = {193-202},

title = {Vector-valued ergodic theorems for multiparameter additive processes},

url = {http://eudml.org/doc/210634},

volume = {79},

year = {1999},

}

TY - JOUR

AU - Sato, Ryotaro

TI - Vector-valued ergodic theorems for multiparameter additive processes

JO - Colloquium Mathematicae

PY - 1999

VL - 79

IS - 2

SP - 193

EP - 202

AB - Let X be a reflexive Banach space and (Ω,Σ,μ) be a σ-finite measure space. Let d ≥ 1 be an integer and T=T(u):u=($u_{1}$, ... ,$u_{d})$, $u_{i}$ ≥ 0, 1 ≤ i ≤ d be a strongly measurable d-parameter semigroup of linear contractions on $L_{1}$((Ω,Σ,μ);X). We assume that to each T(u) there corresponds a positive linear contraction P(u) defined on $L_{1}$((Ω,Σ,μ);ℝ) with the property that ∥ T(u)f(ω)∥ ≤ P(u)∥f(·)∥(ω) almost everywhere on Ω for all f ∈ $L_{1}$((Ω,Σ,μ);X). We then prove stochastic and pointwise ergodic theorems for a d-parameter bounded additive process F in $L_{1}$((Ω,Σ,μ);X) with respect to the semigroup T.

LA - eng

KW - multiparameter dynamical system; ergodic theorem; multiparameter additive processes

UR - http://eudml.org/doc/210634

ER -

## References

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- [3] A. M. Garsia, Topics in Almost Everywhere Convergence, Markham, Chicago, 1970. Zbl0198.38401
- [4] S. Hasegawa and R. Sato, On d-parameter pointwise ergodic theorems in ${L}_{1}$, Proc. Amer. Math. Soc. 123 (1995), 3455-3465. Zbl0849.47007
- [5] S. Hasegawa and R. Sato, On a d-parameter ergodic theorem for continuous semigroups of operators satisfying norm conditions, Comment. Math. Univ. Carolin. 38 (1997), 453-462. Zbl0937.47009
- [6] S. Hasegawa, R. Sato and S. Tsurumi, Vector valued ergodic theorems for a one-parameter semigroup of linear operators, Tôhoku Math. J. 30 (1978), 95-106. Zbl0377.47008
- [7] U. Krengel, Ergodic Theorems, de Gruyter, Berlin, 1985.
- [8] R. Sato, Vector valued differentiation theorems for multiparameter additive processes in ${L}_{p}$ spaces, Positivity 2 (1998), 1-18. Zbl0915.47012

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