| Title:
|
Ergodic properties of contraction semigroups in $L_p$, $1<p<\infty$ (English) |
| Author:
|
Sato, Ryotaro |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
35 |
| Issue:
|
2 |
| Year:
|
1994 |
| Pages:
|
337-346 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\{T(t):t>0\}$ 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 $|T(t)f|\leq P(t)|f|$ for all $f\in L_p$, then there exists a strongly continuous semigroup $\{S(t):t>0\}$ of positive linear contractions in $L_p$ such that $|T(t)f|\leq S(t)|f|$ 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. (English) |
| Keyword:
|
contraction semigroup |
| Keyword:
|
semigroup modulus |
| Keyword:
|
majorant |
| Keyword:
|
pointwise ergodic \newline theorem |
| Keyword:
|
pointwise local ergodic theorem |
| MSC:
|
47A35 |
| MSC:
|
47B38 |
| MSC:
|
47D03 |
| MSC:
|
47D06 |
| idZBL:
|
Zbl 0814.47010 |
| idMR:
|
MR1286580 |
| . |
| Date available:
|
2009-01-08T18:11:26Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118672 |
| . |
| Reference:
|
[1] Akcoglu M.A.: A pointwise ergodic theorem in $L_p$-spaces.Canad. J. Math. 27 (1975), 1075-1082. Zbl 0326.47005, MR 0396901 |
| Reference:
|
[2] Akcoglu M.A., Krengel U.: Two examples of local ergodic divergence.Israel J. Math. 33 (1979), 225-230. Zbl 0441.47007, MR 0571531 |
| Reference:
|
[3] Dunford N., Schwartz J.T.: Linear Operators. Part I: General Theory.Interscience Publishers, New York, 1958. Zbl 0635.47001, MR 1009162 |
| Reference:
|
[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. MR 0823078 |
| Reference:
|
[5] Krengel U.: Ergodic Theorems.Walter de Gruyter, Berlin, 1985. Zbl 0649.47042, MR 0797411 |
| Reference:
|
[6] Sato R.: A note on a local ergodic theorem.Comment. Math. Univ. Carolinae 16 (1975), 1-11. Zbl 0296.28019, MR 0365182 |
| Reference:
|
[7] Sato R.: Contraction semigroups in Lebesgue space.Pacific J. Math. 78 (1978), 251-259. Zbl 0363.47021, MR 0513298 |
| Reference:
|
[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. MR 0308848 |
| . |
