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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:XX is said to be hypercyclic if there exists a vector xX, called hypercyclic for T, such that the orbit Orb(T,x)={Tnx:nN} is dense in X. T is hereditarily hypercyclic if and only if TT is hypercyclic on X×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 xX 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 xX 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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