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Korean Mathematical Society
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ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2008; 45(4): 903-921

Printed July 1, 2008

Copyright © The Korean Mathematical Society.

Hypercyclicity on invariant subspaces

Henrik Petersson

School of Mathematical Sciences Chalmers

Abstract

A continuous linear operator $T: \mathcal X \to \mathcal X $ is called hypercyclic if there exists an $x\in \mathcal X$ such that the orbit $\{ T^n x\}_{n\geq 0}$ is dense. We consider the problem: given an operator $T: \mathcal X \to \mathcal X$, hypercyclic or not, is the restriction $T|_\mathcal Y$ to some closed invariant subspace $\mathcal Y \subset \mathcal X$ hypercyclic? In particular, it is well-known that any non-constant partial differential operator $p(D)$ on $H(\mathbb C^d)$ (entire functions) is hypercyclic. Now, if $q(D)$ is another such operator, $p(D)$ maps $\ker q(D)$ invariantly (by commutativity), and we obtain a necessary and sufficient condition on $p$ and $q$ in order that the restriction $p(D):\ker q(D)\to \ker q(D)$ is hypercyclic. We also study hypercyclicity for other types of operators on subspaces of $H(\mathbb C^d)$.

Keywords: hypercyclic, restriction, extension, invariant subspace

MSC numbers: 47A15, 47A16, 47B38, 32A70, 35A35

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