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By Frédéric Bayart, Étienne Matheron

ISBN-10: 0521514967

ISBN-13: 9780521514965

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1 ⎠ 0 ... 0 0 We have to show that, given (u, v) ∈ KN × KN , one can find a sequence (xn )n∈N in K2N such that xn → u 0 and (I + B)n xn → v . 0 A straightforward computation shows that (I + B)n = Kn 0 Ln Kn for any n ≥ 2N , where ⎛ ⎜ ⎜ Kn = ⎜ ⎜ ⎝ 1 n 1 0 .. 1 .. 0 ... .. .. 0 n N −1 .. 1 Mixing operators and ⎛ ⎜ ⎜ ⎜ Ln = ⎜ ⎜ ⎝ n N n N +1 n N −1 n N .. .. n 1 ... 35 n 2N −1 ... . . . n N +1 n N n N −1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ Thus, we have to find two sequences (un ), (un ) in KN such that un → u and un → 0; Kn un + Ln un → v and Kn un → 0.

Godefroy, the proof yields in fact the following more general result: if X is a separable Fr´echet space and if G ⊂ X is a Gδ set containing a dense linear subspace of X then G is homeomorphic to X. 4 Three examples We conclude this chapter by studying hypercyclicity in three classical families of operators. Our aim is to illustrate some of the beautiful features of concrete operator theory that appear in linear dynamics. 1 Weighted shifts In this subsection we characterize the hypercyclicity and the supercyclicity of a weighted backward shift in terms of its weight sequence.

An operator T ∈ L(X) is said to be (topologically) mixing if the following property holds true: for any pair (U, V ) of non-empty open subsets of X, one can find an N ∈ N such that T n (U ) ∩ V = ∅ for all n ≥ N . 31 32 Hypercyclicity everywhere By its very definition, the mixing property is a strong form of topological transitivity; in particular, mixing operators are hypercyclic if the underlying space X is a separable F -space. In fact, in this case it is easy to see that an operator T ∈ L(X) is mixing if and only if it is hereditarily hypercyclic, which means that for any infinite set N ⊂ N the family {T n ; n ∈ N} is universal.

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Dynamics of Linear Operators by Frédéric Bayart, Étienne Matheron


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