Abstract
In this chapter we obtain, among other things, the Ansari–Bernal theorem that every infinite-dimensional separable Banach space supports a hypercyclic operator. In contrast, some infinite-dimensional separable Banach spaces do not support any chaotic operator. We also discuss here the richness of the set of hypercyclic operators in two ways: it forms a dense set in the space of all operators when endowed with the strong operator topology; and it is shown that any linearly independent sequence of vectors appears as the orbit under a hypercyclic operator.
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© 2011 Springer-Verlag London Limited
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Grosse-Erdmann, KG., Peris Manguillot, A. (2011). Existence of hypercyclic operators. In: Linear Chaos. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-2170-1_8
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DOI: https://doi.org/10.1007/978-1-4471-2170-1_8
Publisher Name: Springer, London
Print ISBN: 978-1-4471-2169-5
Online ISBN: 978-1-4471-2170-1
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