Dense range perturbations of hypercyclic operators

Luis Bernal-Gonzalez¹

¹ Departamento de Análisis Matemático Facultad de Matemáticas Universidad de Sevilla Apdo. 1160, Avenida Reina Mercedes 41080 Sevilla, Spain

Colloquium Mathematicum, Tome 91 (2002) no. 2, pp. 283-292

Résumé

We show that if $(T_n)$ is a hypercyclic sequence of linear operators on a locally convex space and $(S_n)$ is a sequence of linear operators such that the image of each orbit under every linear functional is non-dense then the sequence $(T_n + S_n)$ has dense range. Furthermore, it is proved that if $T,S$ are commuting linear operators in such a way that $T$ is hypercyclic and all orbits under $S$ satisfy the above non-denseness property then $T - S$ has dense range. Corresponding statements for operators and sequences which are hypercyclic in a weaker sense are shown. Our results extend and improve a result on denseness due to C. Kitai.

Détail

DOI : 10.4064/cm91-2-7

Keywords: hypercyclic sequence linear operators locally convex space sequence linear operators image each orbit under every linear functional non dense sequence has dense range furthermore proved commuting linear operators hypercyclic orbits under satisfy above non denseness property has dense range corresponding statements operators sequences which hypercyclic weaker sense shown results extend improve result denseness due kitai