Journal of Operator Theory

Volume 80, Issue 1, Summer 2018 pp. 3-24.

Topologies for which every nonzero vector is hypercyclic

Authors: Henrik Petersson
Author institution: Hvitfeldtska gymnasiet, Gothenburg, 41132, Sweden

Summary: An operator $T:X\to X$ is said to be hypercyclic if there exists a vector $x\in X$, called hypercyclic for $T$, such that the orbit $\mathrm{Orb} (T,x)=\{T^n x :n\in \mathbb{N}\}$ is dense in $X$. $T$ is hereditarily hypercyclic if and only if $T\oplus T$ is hypercyclic on $X\times X$. We show that if $T$ is a hereditarily hypercyclic operator on a Banach space $X$, then there exist separated locally convex topologies on $X^*$ for which every nonzero vector $x^* \in X^*$ is hypercyclic for $T^*$, and thus for which $T^*$ lacks nontrivial closed invariant subsets. We obtain in this way a link between properties of these topologies and the structure of hypercyclic vectors for $T$. In the same way, given that $T^*$ is hereditarily hypercyclic, we can construct separated locally convex topologies on $X$ where any nonzero vector $x \in X$ is hypercyclic for $T$. We introduce the notion of a nondegenerating hypercyclic vector manifold for an operator; such manifolds play a central role here, but these structures are also of independent interest.

DOI: http://dx.doi.org/10.7900/jot.2017may15.2178
Keywords: hypercyclicity, hypercyclicity criterion, invariant subset, multiplier, nondegenerating hypercyclic vector manifold

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