Hypercyclic operators failing the hypercyclicity criterion on classical Banach spaces.

By Fr'ed'eric Bayart, É. Matheron. 2007

86 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b3f1399c3d01473fedad762598102abfb19a821b

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In Algebra and Number Theory

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Linear chaos and frequent hypercyclicity (2014, background): “More precisely, Bayart and Matheron consider an increasing sequence of nonnegative integers (bn)n≥0 with b0 = 0 and an operator T of the form Tek = { wk+1ek+1 if k ∈ [bn, bn+1 − 1[ εnebn + fn if k = bn − 1 where fn = ∑bn−1 k=0 fn,kek and εn > 0.”
Recurrent Linear Operators (2013, background): “question that was only recently settled, in the negative, by de la Rosa and Read [16], and also by Bayart and Matheron, [7], in classical Banach spaces.”
Subspace hypercyclicity (2010, background): “Furthermore, Bayart and Matheron showed in [2] that the equivalence fails on classical Banach spaces, and even on Hilbert space.”
Hypercyclic weighted translations on groups (2011, result): “Hypercyclic operators have been studied by many authors; we refer to [2, 3, 5, 7, 9, 11, 12, 16, 17, 18] for some recent works.”
Linear subsets of nonlinear sets in topological vector spaces (2013, background): “Finally, de la Rosa and Read [130] settled the problem in the negative, and several examples of hypercyclic operators T defined on classical spaces X and not satisfying the HCC have been provided by Bayart and Matheron in [51].”
Szemeredi's theorem, frequent hypercyclicity and multiple recurrence (2010, background): “This was a long standing question that was solved in [18] and in a more general context in [5].”
Linear dynamical systems on Hilbert spaces: typical properties and explicit examples (2017, methodology): “For example, the fact that topological weak mixing is a typical property seems to prevent the use of “soft” arguments for establishing the existence of hypercyclic operators which are not weakly mixing; and indeed, the proofs in [19] and [7] rely on rather technical arguments.”