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Spectral theory and hypercyclic subspaces

Authors: Fernando León-Saavedra and Alfonso Montes-Rodríguez
Journal: Trans. Amer. Math. Soc. 353 (2001), 247-267
MSC (2000): Primary 47A16, 47A53; Secondary 47B37
Published electronically: September 13, 2000
MathSciNet review: 1783790
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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.

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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

Alfonso Montes-Rodríguez
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Avenida Reina Mercedes, Apartado 1160, Sevilla 41080, Spain

Keywords: Hypercyclic operator, hypercyclic vector, essential spectrum, essential minimum modulus, bilateral shift, backward shift, multiplier, composition operator, differentiation operator, translation operator
Received by editor(s): April 14, 1997
Published electronically: September 13, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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