On the spectrum of frequently hypercyclic operators

By Stanislav Shkarin. 2008

85 citations recorded by the source corpus.

A bounded linear operator T on a Banach space X is called fre quently hypercyclic if there exists x E X such that the lower density of the set {n E N: T'x E U} is positive for any non-empty open subset U of X. Bayart and Grivaux have raised a question whether there is a frequently hypercyclic operator on any separable infinite dimensional Banach space. We prove that the spectrum of a frequently hypercyclic operator has no isolated points. It follows that there are no frequently hypercyclic operators on all complex and on some real hereditarily indecomposable Banach spaces, which provides a negative answer to the above question.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9aed444707a35e44eafcc794170c878452c196a5

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Linear dynamical systems on Hilbert spaces: typical properties and explicit examples (2017, background): “This notion was introduced by Shkarin in [49].”
Linear chaos and frequent hypercyclicity (2014, methodology): “Two other quantifications of the frequency of visits of an orbit were investigated by Shkarin [32] and by Bès, Peris, Puig and the author [12]: U-frequent hypercyclicity and reiterative hypercyclicity.”
Mixing operators and small subsets of the circle (2011, background): “Is T mixing in the Gaussian sense? (6) On which Banach spaces is it possible to find ergodic operators (in the Gaussian sense) with no eigenvalues? The spaceX should of course not have cotype 2, but this is not enough: for example, if X is hereditarily indecomposable then there are no frequently hyp”
Recurrence properties of hypercyclic operators (2014, background): “Frequently hypercyclic operators were introduced by Bayart and Grivaux [2] while U -frequently hypercyclic operators were introduced by Shkarin [20].”
Upper frequent hypercyclicity and related notions (2016, background): “Now, in [37], Shkarin has introduced the weaker notion of upper (or U -)frequently hypercyclic operators.”
Difference sets and frequently hypercyclic weighted shifts (2013, background): “Of course, one may also investigate the corresponding notion replacing lower by upper density, leading to the following definition introduced in [16].”
Frequently recurrent operators (2020, methodology): “sitive upper Banach density, respectively). An operator that possesses such a vector is called frequently hypercyclic (upper frequently hypercyclic, reiteratively hypercyclic, respectively), see [3], [42], [9], [13], and the textbooks [5], [26]. Note that uniformrecurrence admits no hypercyclic anal”
Subspace hypercyclicity (2010, background): “By a theorem of Shkarin [24], one can construct a hypercyclic operator T with spectrum K0.”
A new class of frequently hypercyclic operators (2010, background): “Frequent hypercyclicity is a much stronger notion than hypercyclicity, and some operators are hypercyclic without being frequently hypercyclic: an example is the Bergman backward shift [3], and then it was proved in [18] that no hypercyclic operator whose spectrum has an isolated point can be freque”