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

Peris, Alfred

Funct. Approximatio, Comment. Math. 59, No. 2, 279-284 (2018).

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.

MSC:

47A16	Cyclic vectors, hypercyclic and chaotic operators
47B37	Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)

Keywords:

hypercyclic operators; sequentially hypercyclic; composition operator; direct sums of sequence space; comparison principle

Citations:

Zbl 1272.47037; Zbl 1150.47005

References:

[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