Ergodicity of commuting multioperators and holomorphic multioperators of multiplication
DOI:
https://doi.org/10.22199/issn.0717-6279-5790Keywords:
ergodic theorem, holomorphic vector-valued functions, commuting multioperators, power bounded, multioperators of multiplicationAbstract
In this paper, the strong ergodic theorems are extended from the case of one bounded operator to the case of commuting multioperators. The authors show that in Grothendieck space with the Dunford-Pettis property, mean ergodic operator, and uniform ergodic operator are the same. We study when multioperators of multiplication on a weighted Banach space of holomorphic multi-functions are power bounded, mean ergodic, or uniformly mean ergodic.
References
E. Barletta and S. Dragomir, “Vector valued holomorphic functions”, Bulletin mathématique de la Société des Sciences Mathématiques de Roumanie Nouvelle Série, vol. 52, no. 100, no. 3, 2009, pp. 211-226, 2009.
K. D. Bierstedt and J. Bonet, “Weighted (LB)-spaces of holomorphic functions: V H(G) = V0H(G) and completeness of V0H(G)”, Journal of Mathematical Analysis and Applications, vol. 323, pp. 747-767, 2006. https://doi.org10.1016/j.jmaa.2005.10.075
K. D. Bierstedt and S. Holtmanns, “An operator representation for weighted spaces of vector valued holomorphic functions”, Results in Mathematics, vol. 36, pp. 9-20, 1999. https://doi.org10.1007/BF03322097
J. Bonet, P. Domański, and M. Lindström, “Pointwise multiplication operators on weighted Banach spaces of analytic functions”, Studia Mathematica, vol. 137, no. 2, pp. 177-194, 1999. [On line]. Available: https://bit.ly/44r8jtG
J. Bonet, and W. Ricker, “Mean ergodicity of multiplication operators in weighted spaces of holomorphic functions”, Archiv der Mathematik, vol. 92, pp. 428-437, 2009. https://doi.org 10.1007/s00013-009-3061-1
M. Chō and M. Takaguchi, “Boundary points of joint numerical ranges”, Pacific Journal of Mathematics, vol. 95, no. 1, pp. 27-35, 1981. https://doi.org10.2140/PJM.1981.95.27
E. Hille, “Remarks on ergodic theorems”, Transactions of the American Mathematical Society, vol. 57, no.2, pp. 246-269, 1945. https://doi.org10.2307/1990205
E. Jorda, “Weighted Vector-Valued Holomorphic Functions on Banach Spaces”, Abstract and Applied Analysis, vol. 2013, 2013. doi. 10.1155/2013/501592
U. Krengel, Ergodic Theorems. Berlin: Walter de Gruyter, 1985. https://doi.org10.1515/9783110844641
M. Lin, “On the uniform ergodic theorem”, Proceedings of the American Mathematical Society, vol. 43, no. 2, pp. 337-340, 1974.
H. P. Lotz, “Tauberian theorems for operators on L∞ and similar spaces”, North-Holland Mathematics Studies, vol. 90, pp. 117-133, 1984. https://doi.org10.1016/S0304-0208(08)71470-1
H. P. Lotz, “Uniform convergence of operators on L∞ and similar spaces”, Mathematische Zeitschrift, vol. 190, no. 2, pp. 207-220, 1985. https://doi.org10.1007/BF01160459
W. Lusky, “On the isomorphism classes of weighted spaces of harmonic and holomorphic functions”, Studia Mathematica, vol. 175, no. 1, pp. 19-45, 2006. https://doi.org10.4064/sm175-1-2
W. Lusky, J. Taskinen, “On weighted spaces of holomorphic functions of several variables”, Israel Journal of Mathematics, vol. 176, no. 1, pp. 381-399, 2010. https://doi.org10.1007/s11856-010-0033-x
M. Mbekhta, F. H. Vasilescu, “Uniformly ergodic multioperators”, Transactions of the American Mathematical Society, vol. 347, no. 5, pp. 1847-1854, 1995.
V. Müller, Spectral theory of linear operators. Operator Theory Advances and Applications, vol. 139. Basel: Birkhäuser, 2003. https://doi.org10.1007/978-3-7643-8265-0
R. K. Singh, J. S. Manhas, “Multiplication operators on weighted spaces of vector-valued continuous functions”, Journal of the Australian Mathematical Society, vol. 50, no. 1, pp. 98-107, 1991. https://doi.org10.1017/S1446788700032584
J. L. Taylor, “The analytic functional calculus for several commuting operators”, Acta Mathematica, vol. 125, pp. 1-38, 1970. https://doi.org10.1007/BF02392329
Published
How to Cite
Issue
Section
Copyright (c) 2023 Abdellah Akrym, Abdeslam El Bakkali, Abdelkhalek Faouzi
This work is licensed under a Creative Commons Attribution 4.0 International License.
-
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.