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→X
is said to be hypercyclic if there exists a vector x∈X, called hypercyclic for T, such that
the orbit Orb(T,x)={Tnx:n∈N} is dense in X. T is hereditarily hypercyclic if and only if T⊕T 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 x∗∈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∈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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