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SEMI-FREDHOLM THEORY: HYPERCYCLIC AND SUPERCYCLIC SUBSPACES

Published online by Cambridge University Press:  01 July 2000

MANUEL GONZÁLEZ
Affiliation:
Departamento de Matemáicas, Facultad de Ciencias, Avendia de los Castros s/n, E-39005, Santander, Spain
FERNANDO LEÓN-SAAVEDRA
Affiliation:
Departamento de Matemáticas, Escuela Superior de Ingenieeería, C/Sacramento 82, E-11003, Cadiz, Spain
ALFONSO MONTES-RODRÍGUEZ
Affiliation:
Departamento de Análisis Matemático, Facultad de Matemáticas, Avenida Reina Mercedes, Apartado 1160, E-41080, Sevilla, Spain
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Abstract

A vector $x$ in a Banach space $\cal B$ is called hypercyclic for a bounded linear operator $T: \cal B \rightarrow \cal B $ if the orbit $\{T^n x : n \geq 1 \}$ is dense in $\cal B$. We prove that if $T$ is hereditarily hypercyclic and its essential spectrum meets 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 also true even if $T$ is not hereditarily hypercyclic. In this way, we improve and extend to Banach spaces a recent result for Hilbert spaces. We also investigate the corresponding problem for supercyclic operators. In this case we obtain a description of the norm closure of the class of all supercyclic operators that have an infinite-dimensional closed subspace of hypercyclic vectors. Moreover, for certain kinds of supercyclic operators we can characterize when they have an infinite-dimensional closed subspace of supercyclic vectors. 1991 Mathematics Subject Classification: 47B38, 47B99.

Type
Research Article
Copyright
2000 London Mathematical Society

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