×

On local ergodic convergence of semi-groups and additive processes. (English) Zbl 0545.47016

Let \(\{T_ t\}_{t>0}\) be a strongly continuous (at \(t>0)\) semigroup of bounded linear operators in the \(L_ p\) space \((1\leq p<\infty)\) of a probability space \((X,\Sigma\),m). The question considered is: under what conditions does \(\lim_{\epsilon \to 0+}\frac{1}{\epsilon}\int^{\epsilon}_{0}T_ tf(x)dt\) exist a.e. for every \(f\in L_ p?\)
Many authors have considered this question with various restrictions on p and \(\{T_ t\}\). Here the author considers positive operators \(T_ t\) on \(L_ 1\) and shows that the above limit exists for every \(f\in L_{\infty}\) and \(\sup_{0<t\leq 1}\| T_ t\|<\infty\) if and only if \(\{T_ t\}\) is strongly continuous at zero (in \(L_ 1).\)
The method used in the proof is then used to prove various other results in this area and some results for n-parameter semigroups.
Reviewer: D.Newton

MSC:

47A35 Ergodic theory of linear operators
47D03 Groups and semigroups of linear operators
Full Text: DOI

References:

[1] Akcoglu, M. A.; Chacon, R. V., A local ratio theorem, Can. J. Math., 22, 545-552 (1970) · Zbl 0201.06603
[2] M. A. Akcoglu and A. del Junco,Differentiation of n-dimensional additive processes, Can. J. Math., to appear. · Zbl 0477.47012
[3] Akcoglu, M. A.; Krengel, U., A differentiation theorem for additive processes, Math. Z., 163, 199-210 (1978) · Zbl 0379.60073 · doi:10.1007/BF01214067
[4] Akcoglu, M. A.; Krengel, U., A differentiation theorem in L_p, Math. Z., 169, 31-40 (1979) · Zbl 0394.47021 · doi:10.1007/BF01214911
[5] Akcoglu, M.; Krengel, U., Two examples of local ergodic divergence, Isr. J. Math., 33, 225-230 (1979) · Zbl 0441.47007
[6] Baxter, J. R.; Chacon, R. V., A local ergodic theorem on L_p, Can. J. Math., 26, 1206-1216 (1974) · Zbl 0285.47007
[7] Hille, E.; Phillips, R. S., Functional Analysis and Semi-Groups (1957), Providence: AMS Colloquium Publications, Providence · Zbl 0078.10004
[8] Kipnis, C., Majoration des semi-groupes de contractions de L_1 et applications, Ann. Inst. Poincaré Sect. B, 10, 369-384 (1974) · Zbl 0324.47004
[9] Krengel, U., A local ergodic theorem, Invent. Math., 6, 329-333 (1969) · Zbl 0165.37402 · doi:10.1007/BF01425423
[10] Krengel, U., A necessary and sufficient condition for the validity of the local ergodic theorem, Springer Lecture Notes in Math, 89, 170-177 (1969) · Zbl 0186.49802
[11] Kubokawa, Y., A general local ergodic theorem, Proc. Jpn. Acad., 48, 461-465 (1972) · Zbl 0254.47013
[12] Kubokawa, Y., A local ergodic theorem for semi-groups on L_p, Tôhoku Math. J., 26, 2, 411-422 (1974) · Zbl 0289.47025
[13] Kubokawa, Y., Ergodic theorems for contraction semi-groups, J. Math. Soc. Jpn., 27, 184-193 (1975) · Zbl 0299.47007
[14] Lin, M., Semi-groups of Markov operators, Boll. Unione Mat. Ital., 6, 4, 20-44 (1972) · Zbl 0276.60071
[15] Ornstein, D. S., The sums of iterates of a positive operator, Advances in Prob. and Related Topics, 2, 87-115 (1970) · Zbl 0321.28013
[16] Sato, R., A note on a local ergodic theorem, Comment. Math. Univ. Carolinae, 16, 1-11 (1975) · Zbl 0296.28019
[17] Sato, R., On a local ergodic theorem, Studia Math, 58, 1-5 (1976) · Zbl 0344.47005
[18] Sato, R., On local ergodic theorems for positive semi-groups, Studia Math, 63, 45-55 (1978) · Zbl 0391.47022
[19] Sato, R., Contraction semi-groups in Lebesgue space, Pac. J. Math., 78, 251-259 (1978) · Zbl 0363.47021
[20] Sato, R., Two local ergodic theorems in L_∞, J. Math. Soc. Jpn., 32, 415-423 (1980) · Zbl 0433.47021 · doi:10.2969/jmsj/03230415
[21] Terrell, T. R., Local ergodic theorems for n-parameter semi-groups of operators, Springer Lecture Notes in Math, 160, 262-278 (1970) · Zbl 0204.45406
[22] Wiener, N., The ergodic theorem, Duke Math. J., 5, 1-18 (1939) · Zbl 0021.23501 · doi:10.1215/S0012-7094-39-00501-6
[23] McGrath, S., On the local ergodic theorems of Krengel, Kubokawa and Terrell, Commun. Math. Univ. Carolinae, 17, 49-59 (1976) · Zbl 0327.28015
[24] Gapoŝkin, V. F., The local ergodic theorem for groups of unitary operators, Mat. Sb., 39, 227-242 (1981) · Zbl 0462.47007 · doi:10.1070/SM1981v039n02ABEH001486
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.