Unimodular eigenvalues, uniformly distributed sequences and linear dynamics

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Abstract

We study increasing sequences of positive integers (nk)k ≥ 1with the following property: every bounded linear operator T acting on a separable Banach (or Hilbert) space with supk ≥ 1{norm of matrix} Tnk{norm of matrix} < ∞ has a countable set of unimodular eigenvalues. Whether this property holds or not depends on the distribution (modulo one) of sequences (nkα)k ≥ 1, α ∈ R, or on the growth of nk + 1/ nk. Counterexamples to some conjectures in linear dynamics are given. For instance, a Hilbert space operator which is frequently hypercyclic, chaotic, but not topologically mixing is constructed. The situation of C0-semigroups is also discussed. © 2006 Elsevier Inc. All rights reserved.

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Badea, C., & Grivaux, S. (2007). Unimodular eigenvalues, uniformly distributed sequences and linear dynamics. Advances in Mathematics, 211(2), 766–793. https://doi.org/10.1016/j.aim.2006.09.010

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