HYPERCYCLICITY ON INVARIANT SUBSPACES

Title & Authors
HYPERCYCLICITY ON INVARIANT SUBSPACES

Abstract
A continuous linear operator $\small{T\;:\;X{\rightarrow}X}$ is called hypercyclic if there exists an $\small{x\;{\in}\;X}$ such that the orbit $\small{{T^nx}_{n{\geq}0}}$ is dense. We consider the problem: given an operator $\small{T\;:\;X{\rightarrow}X}$, hypercyclic or not, is the restriction $\small{T|y}$ to some closed invariant subspace $\small{y{\subset}X}$ hypercyclic? In particular, it is well-known that any non-constant partial differential operator p(D) on $\small{H({\mathbb{C}}^d)}$ (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) $\small{\rightarrow}$ ker q(D) is hypercyclic. We also study hypercyclicity for other types of operators on subspaces of $\small{H({\mathbb{C}}^d)}$.
Keywords
hypercyclic;restriction;extension;invariant subspace;
Language
English
Cited by
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