# Ergodic properties of contraction semigroups in ${L}_{p}$, $1<p<\infty $

Commentationes Mathematicae Universitatis Carolinae (1994)

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

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topSato, 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

top- Akcoglu M.A., A pointwise ergodic theorem in ${L}_{p}$-spaces, Canad. J. Math. 27 (1975), 1075-1082. (1975) Zbl0326.47005MR0396901
- Akcoglu M.A., Krengel U., Two examples of local ergodic divergence, Israel J. Math. 33 (1979), 225-230. (1979) Zbl0441.47007MR0571531
- Dunford N., Schwartz J.T., Linear Operators. Part I: General Theory, Interscience Publishers, New York, 1958. Zbl0635.47001MR1009162
- 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
- Krengel U., Ergodic Theorems, Walter de Gruyter, Berlin, 1985. Zbl0649.47042MR0797411
- Sato R., A note on a local ergodic theorem, Comment. Math. Univ. Carolinae 16 (1975), 1-11. (1975) Zbl0296.28019MR0365182
- Sato R., Contraction semigroups in Lebesgue space, Pacific J. Math. 78 (1978), 251-259. (1978) Zbl0363.47021MR0513298
- 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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