Ergodic properties of contraction semigroups in L p , 1 < p < ∞

Ryotaro Sato

Commentationes Mathematicae Universitatis Carolinae (1994)

  • Volume: 35, Issue: 2, page 337-346
  • ISSN: 0010-2628

Abstract

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Let { T ( t ) : t > 0 } be a strongly continuous semigroup of linear contractions in L p , 1 < p < ∞ , of a σ -finite measure space. In this paper we prove that if there corresponds to each t > 0 a positive linear contraction P ( t ) in L p such that | T ( t ) f | ≤ P ( t ) | f | for all f ∈ L p , then there exists a strongly continuous semigroup { S ( t ) : t > 0 } of positive linear contractions in L p such that | T ( t ) f | ≤ S ( t ) | f | for all t > 0 and f ∈ L p . Using this and Akcoglu’s dominated ergodic theorem for positive linear contractions in L p , we also prove multiparameter pointwise ergodic and local ergodic theorems for such semigroups.

How to cite

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Sato, Ryotaro. "Ergodic properties of contraction semigroups in $L_p$, $1<p<\infty $." Commentationes Mathematicae Universitatis Carolinae 35.2 (1994): 337-346. <http://eudml.org/doc/247630>.

@article{Sato1994,
abstract = {Let $\lbrace T(t):t>0\rbrace $ be a strongly continuous semigroup of linear contractions in $L_p$, $1<p<\infty $, of a $\sigma $-finite measure space. In this paper we prove that if there corresponds to each $t>0$ a positive linear contraction $P(t)$ in $L_p$ such that $|T(t)f|\le P(t)|f|$ for all $f\in L_p$, then there exists a strongly continuous semigroup $\lbrace S(t):t>0\rbrace $ of positive linear contractions in $L_p$ such that $|T(t)f|\le S(t)|f|$ for all $t>0$ and $f\in L_p$. Using this and Akcoglu’s dominated ergodic theorem for positive linear contractions in $L_p$, we also prove multiparameter pointwise ergodic and local ergodic theorems for such semigroups.},
author = {Sato, Ryotaro},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {contraction semigroup; semigroup modulus; majorant; pointwise ergodic theorem; pointwise local ergodic theorem; contraction semigroup; strongly continuous semigroup of linear contractions; strongly continuous semigroup; Akcoglu’s dominated ergodic theorem for positive linear contractions in ; multiparameter pointwise ergodic and local ergodic theorems},
language = {eng},
number = {2},
pages = {337-346},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Ergodic properties of contraction semigroups in $L_p$, $1<p<\infty $},
url = {http://eudml.org/doc/247630},
volume = {35},
year = {1994},
}

TY - JOUR
AU - Sato, Ryotaro
TI - Ergodic properties of contraction semigroups in $L_p$, $1<p<\infty $
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1994
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 35
IS - 2
SP - 337
EP - 346
AB - Let $\lbrace T(t):t>0\rbrace $ be a strongly continuous semigroup of linear contractions in $L_p$, $1<p<\infty $, of a $\sigma $-finite measure space. In this paper we prove that if there corresponds to each $t>0$ a positive linear contraction $P(t)$ in $L_p$ such that $|T(t)f|\le P(t)|f|$ for all $f\in L_p$, then there exists a strongly continuous semigroup $\lbrace S(t):t>0\rbrace $ of positive linear contractions in $L_p$ such that $|T(t)f|\le S(t)|f|$ for all $t>0$ and $f\in L_p$. Using this and Akcoglu’s dominated ergodic theorem for positive linear contractions in $L_p$, we also prove multiparameter pointwise ergodic and local ergodic theorems for such semigroups.
LA - eng
KW - contraction semigroup; semigroup modulus; majorant; pointwise ergodic theorem; pointwise local ergodic theorem; contraction semigroup; strongly continuous semigroup of linear contractions; strongly continuous semigroup; Akcoglu’s dominated ergodic theorem for positive linear contractions in ; multiparameter pointwise ergodic and local ergodic theorems
UR - http://eudml.org/doc/247630
ER -

References

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  1. Akcoglu M.A., A pointwise ergodic theorem in L p -spaces, Canad. J. Math. 27 (1975), 1075-1082. (1975) Zbl0326.47005MR0396901
  2. Akcoglu M.A., Krengel U., Two examples of local ergodic divergence, Israel J. Math. 33 (1979), 225-230. (1979) Zbl0441.47007MR0571531
  3. Dunford N., Schwartz J.T., Linear Operators. Part I: General Theory, Interscience Publishers, New York, 1958. Zbl0635.47001MR1009162
  4. Émilion R., Continuity at zero of semi-groups on L 1 and differentiation of additive processes, Ann. Inst. H. Poincaré Probab. Statist. 21 (1985), 305-312. (1985) MR0823078
  5. Krengel U., Ergodic Theorems, Walter de Gruyter, Berlin, 1985. Zbl0649.47042MR0797411
  6. Sato R., A note on a local ergodic theorem, Comment. Math. Univ. Carolinae 16 (1975), 1-11. (1975) Zbl0296.28019MR0365182
  7. Sato R., Contraction semigroups in Lebesgue space, Pacific J. Math. 78 (1978), 251-259. (1978) Zbl0363.47021MR0513298
  8. Starr N., Majorizing operators between L p spaces and an operator extension of Lebesgue’s dominated convergence theorem, Math. Scand. 28 (1971), 91-104. (1971) MR0308848

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