Unimodular eigenvalues, uniformly distributed sequences and linear dynamics

  • Badea C
  • Grivaux S
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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.

Author-supplied keywords

  • Distribution modulo one
  • Ergodicity and mixing
  • Hypercyclic operators
  • One-parameter semigroups
  • Power-bounded operators
  • Unimodular point spectrum

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Authors

  • Catalin Badea

  • Sophie Grivaux

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