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HYPONORMAL OPERATORS, WEIGHTED SHIFTS AND WEAK FORMS OF SUPERCYCLICITY

Published online by Cambridge University Press:  02 February 2006

Frédéric Bayart  and
Etienne Matheron
Frédéric Bayart
Affiliation:
Laboratoire Bordelais d’Analyse et de Géométrie, UMR 5467, Université Bordeaux 1, 351 Cours de la Libération, 33405 Talence Cedex, France (bayart@math.u-bordeaux1.fr; matheron@math.u-bordeaux1.fr)
Etienne Matheron
Affiliation:
Laboratoire Bordelais d’Analyse et de Géométrie, UMR 5467, Université Bordeaux 1, 351 Cours de la Libération, 33405 Talence Cedex, France (bayart@math.u-bordeaux1.fr; matheron@math.u-bordeaux1.fr)
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Abstract

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An operator $T$ on a Banach space $X$ is said to be weakly supercyclic (respectively $N$-supercyclic) if there exists a one-dimensional (respectively $N$-dimensional) subspace of $X$ whose orbit under $T$ is weakly dense (respectively norm dense) in $X$. We show that a weakly supercyclic hyponormal operator is necessarily a multiple of a unitary operator, and we give an example of a weakly supercyclic unitary operator. On the other hand, we show that hyponormal operators are never $N$-supercyclic. Finally, we characterize $N$-supercyclic weighted shifts.

Keywords

Primary 47A1647B2043A46hypercyclicsupercyclichyponormal operatorsweighted shiftsKronecker sets
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 49 , Issue 1 , February 2006 , pp. 1 - 15
Copyright
Copyright © Edinburgh Mathematical Society 2006