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Frequently hypercyclic operators and vectors

Published online by Cambridge University Press:  09 March 2007

A. BONILLA  and
K.-G. GROSSE-ERDMANN
A. BONILLA
Affiliation:
Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna, Tenerife, Spain (e-mail: [email protected])
K.-G. GROSSE-ERDMANN
Affiliation:
Fakultät für Mathematik und Informatik, FernUniversität Hagen, 58084 Hagen, Germany (e-mail: [email protected])
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Abstract

We study frequently hypercyclic operators, a natural new concept in hypercyclicity that was recently introduced by F. Bayart and S. Grivaux. We derive a strengthened version of their Frequent Hypercyclicity Criterion, which allows us to obtain examples of frequently hypercyclic operators in a straightforward way. Moreover, Bayart and Grivaux have noted that the frequent hypercyclicity setting differs from general hypercyclicity in that the set of frequently hypercyclic vectors need not be residual. We show here that, under weak assumptions, this set is only of first category. Motivated by this we study the question of whether one may write every vector in the underlying space as the sum of two frequently hypercyclic vectors. This investigation leads us to the introduction of a new notion, that of Runge transitivity.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 2 , April 2007 , pp. 383 - 404
Copyright
2007 Cambridge University Press

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