A new class of frequently hypercyclic operators

By S. Grivaux. 2010

52 citations recorded by the source corpus.

We study a hypercyclicity property of linear dynamical systems: a bounded linear operator T acting on a separable infinite-dimensional Banach space X is said to be hypercyclic if there exists a vector x in X such that {T^{n}x : n>0} is dense in X, and frequently hypercyclic if there exists x in X such that for any non empty open subset U of X, the set {n>0 ; T^n x \in U} has positive lower density. We prove that if T is a bounded operator on X which has "sufficiently many" eigenvectors associated to eigenvalues of modulus 1 in the sense that these eigenvectors are perfectly spanning, then T is automatically frequently hypercyclic.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9f7ccc5c18ebed8b5f37caea988cd7f5f9158672

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Recurrent Linear Operators (2013, background): “In fact, in this case, T is frequently hypercyclic; see [5,6,29,30].”
Difference sets and frequently hypercyclic weighted shifts (2013, unknown): “This notion has then been investigated by several authors; see, for instance, [3, 5, 8, 10, 11, 16].”
Linear dynamical systems on Hilbert spaces: typical properties and explicit examples (2017, background): “We now proceed as in [28] or [30], and replace each coefficient bs by a certain sum of scalars of the form ∑N−1 j=0 cs+jQ with ∑N−1 j=0 |cs+jQ|(2) sufficiently small.”
Invariant measures for frequently hypercyclic operators (2014, methodology): “Actually, the result is not stated exactly in this way in [16].”
Mixing operators and small subsets of the circle (2011, background): “when the transformation T is a continuous linear operator acting on a topological vector space X ([25], [8], [7], [14], [28]).”
Linear chaos and frequent hypercyclicity (2014, background): “[4] F. Bayart and S. Grivaux, Frequently hypercyclic operators, Trans.”