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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Spectral theory and hypercyclic subspaces
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by Fernando León-Saavedra and Alfonso Montes-Rodríguez PDF
Trans. Amer. Math. Soc. 353 (2001), 247-267 Request permission

Abstract:

A vector $x$ in a Hilbert space $\mathcal {H}$ is called hypercyclic for a bounded operator $T: \mathcal {H} \rightarrow \mathcal {H}$ if the orbit $\{T^{n} x : n \geq 1 \}$ is dense in $\mathcal {H}$. Our main result states that if $T$ satisfies the Hypercyclicity Criterion and the essential spectrum intersects the closed unit disk, then there is an infinite-dimensional closed subspace consisting, except for zero, entirely of hypercyclic vectors for $T$. The converse is true even if $T$ is a hypercyclic operator which does not satisfy the Hypercyclicity Criterion. As a consequence, other characterizations are obtained for an operator $T$ to have an infinite-dimensional closed subspace of hypercyclic vectors. These results apply to most of the hypercyclic operators that have appeared in the literature. In particular, they apply to bilateral and backward weighted shifts, perturbations of the identity by backward weighted shifts, multiplication operators and composition operators. The main result also applies to the differentiation operator and the translation operator $T:f(z)\rightarrow f(z+1)$ defined on certain Hilbert spaces consisting of entire functions. We also obtain a spectral characterization of the norm-closure of the class of hypercyclic operators which have an infinite-dimensional closed subspace of hypercyclic vectors.
References
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Additional Information
  • Fernando León-Saavedra
  • Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Avenida Reina Mercedes, Apartado 1160, Sevilla 41080, Spain
  • Email: fleon@cica.es
  • Alfonso Montes-Rodríguez
  • Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Avenida Reina Mercedes, Apartado 1160, Sevilla 41080, Spain
  • Email: amontes@cica.es
  • Received by editor(s): April 14, 1997
  • Published electronically: September 13, 2000
  • © Copyright 2000 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 353 (2001), 247-267
  • MSC (2000): Primary 47A16, 47A53; Secondary 47B37
  • DOI: https://doi.org/10.1090/S0002-9947-00-02743-4
  • MathSciNet review: 1783790