Kitai's Criterion for composition operators

[en] We present a general and natural framework to study the dynamics of composition operators on spaces of measurable functions, in which we then reconsider the characterizations for hypercyclic and mixing composition operators obtained by Bayart, Darji and Pires in 2018. We show that the notions of hypercyclicity and weak mixing coincide in this context and, if the system is dissipative, the recurrent composition operators agree with the hypercyclic ones. We also give a characterization for invertible composition operators satisfying Kitai's Criterion, and we construct an example of a mixing composition operator not satisfying Kitai's Criterion. For invertible dissipative systems with bounded distortion we show that composition operators satisfying Kitai's Criterion coincide with the mixing operators.

Research center :

CREMMI - Modélisation mathématique et informatique

Disciplines :

Mathematics

Author, co-author :

Gomes, Daniel ; Departamento de Matemática, Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, Campinas, Brazil

Grosse-Erdmann, Karl ; Université de Mons - UMONS > Faculté des Sciences > Service d'Analyse fonctionnelle

Language :

English

Title :

Kitai's Criterion for composition operators

Publication date :

15 July 2025

Journal title :

Journal of Mathematical Analysis and Applications

ISSN :

0022-247X

eISSN :

1096-0813

Publisher :

Academic Press Inc.

Volume :

547

Issue :

Pages :

129347

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://api.elsevier.com/content/article/PII:S0022247X25001283?httpAccept=text/xml

Research unit :

Mathematical Analysis

Research institute :

Complexys

Funders :

Fonds de la Recherche Scientific
Fundação de Amparo à Pesquisa do Estado de São Paulo

Funding text :

The first author was supported by the S\u00E3o Paulo Research Foundation (FAPESP), grants 2021/02672-2 and 2023/03661-0; the second author was supported by the Fonds de la Recherche Scientifique - FNRS under grant n o CDR J.0078.21.

Available on ORBi UMONS :

since 26 February 2025

Statistics

Number of views

13 (3 by UMONS)

Number of downloads

1 (1 by UMONS)

More statistics

Scopus citations^®

Scopus citations^®
without self-citations

OpenCitations

OpenAlex citations

Bibliography

Bayart, F., Darji, U.B., Pires, B., Topological transitivity and mixing of composition operators. J. Math. Anal. Appl. 465 (2018), 125–139.
Bayart, F., Matheron, É., Dynamics of Linear Operators. 2009, Cambridge University Press, Cambridge.
Birkhoff, G.D., Démonstration d'un théorème élémentaire sur les fonctions entières. C. R. Acad. Sci. 189 (1929), 473–475.
Bogachev, V.I., Measure Theory, vol. I. 2007, Springer-Verlag, Berlin.
Costakis, G., Manoussos, A., Parissis, I., Recurrent linear operators. Complex Anal. Oper. Theory 8 (2014), 1601–1643.
D'Aniello, E., Darji, U.B., Maiuriello, M., Shift-like operators on Lp(X). J. Math. Anal. Appl., 515, 2022, 126393.
D'Aniello, E., Maiuriello, M., Seoane-Sepúlveda, J.B., The interplay between recurrence and hypercyclicity in dissipative contexts. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. RACSAM, 118, 2024, 30.
Darji, U.B., Pires, B., Chaos and frequent hypercyclicity for composition operators. Proc. Edinb. Math. Soc. (2) 64 (2021), 513–531.
Fréchet, M., Sur divers modes de convergence d'une suite de fonctions d'une variable. Bull. Calcutta Math. Soc. 11 (1919–1920), 187–206.
Gethner, R.M., Shapiro, J.H., Universal vectors for operators on spaces of holomorphic functions. Proc. Am. Math. Soc. 100 (1987), 281–288.
Grivaux, S., Hypercyclic operators, mixing operators, and the bounded steps problem. J. Oper. Theory 54 (2005), 147–168.
Grosse-Erdmann, K.-G., Hypercyclic and chaotic weighted shifts. Stud. Math. 139 (2000), 47–68.
Grosse-Erdmann, K.-G., Manguillot, A. Peris, Linear Chaos. 2011, Springer, London.
Kalmes, T., Hypercyclic, mixing, and chaotic C0-semigroups induced by semiflows. Ergod. Theory Dyn. Syst. 27 (2007), 1599–1631.
Kalton, N.J., Peck, N.T., Roberts, J.W., An F-Space Sampler. 1984, Cambridge University Press, Cambridge.
Kitai, C., Invariant closed sets for linear operators. Ph.D. thesis, 1982, University of Toronto, Toronto.
Menet, Q., Papathanasiou, D., Dynamics of weighted shifts on ℓp-sums and c0-sums. arXiv:2408.11153.
Peris, A., Hypercyclicity criteria and the Mittag-Leffler theorem. Bull. Soc. R. Sci. Liège 70 (2001), 365–371.
Royden, H.L., Real Analysis. third edition, 1988, Macmillan Publishing Company, New York.
Singh, R.K., Manhas, J.S., Composition Operators on Function Spaces. 1993, North-Holland Publishing Co., Amsterdam.
Shapiro, J.H., Notes on the dynamics of linear operators. unpublished lecture notes, available at www.math.msu.edu/~shapiro, 2001.
Whitley, R., Normal and quasinormal composition operators. Proc. Am. Math. Soc. 70 (1978), 114–118.
Wilansky, A., Summability Through Functional Analysis. 1984, North-Holland Publishing Co., Amsterdam.