Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Existence of ergodic retraction
for noncommutative semigroups in Banach spaces


Author: Osamu Kada
Journal: Proc. Amer. Math. Soc. 127 (1999), 3013-3020
MSC (1991): Primary 47A35, 47H09, 47H20
DOI: https://doi.org/10.1090/S0002-9939-99-04343-9
Published electronically: May 4, 1999
MathSciNet review: 1627160
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the existence of ergodic retraction for a noncommutative semigroup which is right Eberlein-weakly almost periodic in a uniformly convex Banach space.


References [Enhancements On Off] (What's this?)

  • [1] J. B. Baillon, Un théorèm de type ergodic pour les contractions non linéaires dans un espace de Hilbert, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), 1511-1514. MR 51:11205
  • [2] -, Quelques propiétés de convergence asymptotic pour les semigroupes de contractions impaires, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), 75-78. MR 55:1153
  • [3] J. F. Berglund, H. D. Junghenn and P. Milnes, Analysis on semigroups, John Wiley & Sons. MR 91b:43001
  • [4] R. E. Bruck, On the almost-convergence of iterates of a nonexpansive mapping in Hilbert space and the structure of the weak $\omega$-limit set, Israel J. Math. 29 (1978), 1-16. MR 58:2474
  • [5] -, On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces, Israel J. Math. 38 (1979), 304-314. MR 82h:47051
  • [6] A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Vol. 1, American Mathematical Society, 1961. MR 24:A2627
  • [7] M. M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509-544. MR 19:1067c
  • [8] -, Fixed point theorem for compact convex sets, Illinois J. Math. 5 (1961), 585-590. MR 25:1547
  • [9] W. F. Eberlein, Abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc. 67 (1949), 217-240. MR 12:112a
  • [10] S. Goldberg and P. Irwin, Weakly almost periodic vector-valued functions, Diss. Math. 157 (1979), 46 pp. MR 80e:43010
  • [11] N. Hirano, K. Kido and W. Takahashi, Asymptotic behavior of commutative semigroups of nonexpansive mappings in Banach spaces, Nonlinear Analysis 10 (1986), 229-249. MR 87d:47078
  • [12] -, Nonexpansive retractions and nonlinear ergodic theorems in Banach spaces, Nonlinear Analysis 12 (1988), 1269-1281. MR 90a:47138
  • [13] O. Kada, Strong ergodic theorems for commutative semigroups of operators, (submitted).
  • [14] -, Ergodic theorems for noncommutative semigroups of operators, preprint.
  • [15] -, Strong ergodic theorems for noncommutative semigroups in Banach spaces, preprint.
  • [16] O. Kada, A. Lau and W. Takahashi, Asymptotically invariant net and fixed point set for semigroup of nonexpansive mappings, Nonlinear Anal. 29 (1997), 539-550. MR 98d:47156
  • [17] O. Kada and W. Takahashi, Nonlinear ergodic theorems for almost nonexpansive curves over commutative semigroups, Topological Methods in Nonlinear Analysis 5 (1995), 305-324. MR 97f:47053
  • [18] -, Strong convergence and nonlinear ergodic theorems for commutative semigroups of nonexpansive mappings, Nonlinear Analysis 28 (1997), 495-511. MR 97j:47075
  • [19] J. L. Kelley and I. Namioka, Linear topological spaces, Van Nostrand, 1963. MR 29:3851
  • [20] K. Kido and W. Takahashi, Mean ergodic theorems for semigroups of linear operators, J. Math. Anal. Appl. 103 (1984), 387-394. MR 86g:47052
  • [21] A. T. Lau and W. Takahashi, Invariant means and fixed point properties for nonexpansive representations of topological semigroups, Topological Methods in Nonlinear Analysis 5 (1995), 39-57. MR 96i:47101
  • [22] P. Milnes, On vector-valued weakly almost periodic functions, J. London Math. Soc. (2) 22 (1980), 467-472. MR 82i:43007
  • [23] I. Namioka, Følner's condition for amenable semigroups, Math. Scand. 15 (1964), 18-28. MR 31:5062
  • [24] W. M. Ruess and W. H. Summers, Weak almost periodicity and the strong ergodic limit theorem for contraction semigroups, Israel J. Math. 64 (1988), 139-157. MR 90c:47118
  • [25] -, Integration of asymptotically almost periodic functions and weak asymptotic almost periodicity, Diss. Math. 279 (1989). MR 90d:46056
  • [26] -, Weakly almost periodic semigroups of operators, Pacific J. Math. 143 (1990), 175-193. MR 91b:47142
  • [27] -, Ergodic theorems for semigroups of operators, Proc. Amer. Math. Soc. 114 (1992), 423-432. MR 92e:47016
  • [28] H. H. Schaefer, Topological vector spaces, Springer-Verlag, 1971. MR 49:7722
  • [29] W. Takahashi, A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space, Proc. Amer. Math. Soc. 81 (1981), 253-256. MR 82f:47079
  • [30] -, Nonlinear Functional Analysis (Japanese), Kindai-kagakusha, Japan, 1988.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47A35, 47H09, 47H20

Retrieve articles in all journals with MSC (1991): 47A35, 47H09, 47H20


Additional Information

Osamu Kada
Affiliation: Department of Information Sciences, Tokyo Institute of Technology, Oh-Okayama, Meguro-ku, Tokyo 152, Japan
Email: kada@math.sci.ynu.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-99-04343-9
Received by editor(s): January 22, 1996
Received by editor(s) in revised form: July 18, 1997, and January 9, 1998
Published electronically: May 4, 1999
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society