A hypercyclicity criterion for non-metrizable topological vector spaces. (English) Zbl 1475.47010

Summary: We provide a sufficient condition for an operator \(T\) on a non-metrizable and sequentially separable topological vector space \(X\) to be sequentially hypercyclic. This condition is applied to some particular examples, namely, a composition operator on the space of real analytic functions on \(\,]0,1[\,\), which solves two problems of J. Bonet and P. Domański [Math. Proc. Camb. Philos. Soc. 153, No. 3, 489–503 (2012; Zbl 1272.47037)], and the “snake shift” constructed in [J. Bonet et al., Bull. Lond. Math. Soc. 37, No. 2, 254–264 (2005; Zbl 1150.47005)] on direct sums of sequence spaces. The two examples have in common that they do not admit a densely embedded F-space \(Y\) for which the operator restricted to \(Y\) is continuous and hypercyclic, i.e., the hypercyclicity of these operators cannot be a consequence of the comparison principle with hypercyclic operators on F-spaces.


47A16 Cyclic vectors, hypercyclic and chaotic operators
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
Full Text: DOI arXiv Euclid


[1] F. Bayart and É. Matheron, Cambridge Tracts in Mathematics, Vol. 179, Cambridge University Press, Cambridge 2009.
[2] G. Belitskii and Y. Lyubich, The real analytic solutions of the Abel functional equation, Studia Math. 134 (1999), 135–141. · Zbl 0924.39012
[3] J. Bonet and P. Domański, Hypercyclic composition operators on spaces of real analytic functions, Math. Proc. Cambridge Philos. Soc. 153 (2012), 489–503. · Zbl 1272.47037
[4] J. Bonet and P. Domański, Abel’s functional equation and eigenvalues of composition operators on spaces of real analytic functions, Integral Equations Operator Theory 81 (2015), 455–482. · Zbl 1331.47038
[5] J. Bonet, L. Frerick, A. Peris, and J. Wengenroth, Transitive and hypercyclic operators on locally convex spaces, Bull. London Math. Soc. 37 (2005), 254–264. · Zbl 1150.47005
[6] P. Domański and C.,D. Kariksiz, Eigenvalues and dynamical properties of weighted backward shifts on the space of real analytic functions, Studia Math. (to appear).
[7] P. Domański and D. Vogt, The space of real analytic functions has no basis, Studia Math. 142 (2000), 187–200. · Zbl 0990.46015
[8] K.,G. Grosse-Erdmann and A. Peris, Linear Chaos, Universitext, Springer-Verlag London Ltd., London, 2011. · Zbl 1246.47004
[9] S. Shkarin, Hypercyclic operators on topological vector spaces, J. Lond. Math. Soc. 86 (2012), 195–213. · Zbl 1263.47010
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.