Supercyclicity and weighted shifts

Héctor Salas

Studia Mathematica (1999)

  • Volume: 135, Issue: 1, page 55-74
  • ISSN: 0039-3223

Abstract

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An operator (linear and continuous) in a Fréchet space is hypercyclic if there exists a vector whose orbit under the operator is dense. If the scalar multiples of the elements in the orbit are dense, the operator is supercyclic. We give, for Fréchet space operators, a Supercyclicity Criterion reminiscent of the Hypercyclicity Criterion. We characterize the supercyclic bilateral weighted shifts in terms of their weight sequences. As a consequence, we show that a bilateral weighted shift is supercyclic if and only if it satisfies the Supercyclicity Criterion. We exhibit two supercyclic, irreducible Hilbert space operators which are C*-isomorphic, but one is hypercyclic and the other is not. We prove that a Banach space operator which satisfies a version of the Supercyclicity Criterion, and has zero in its left essential spectrum, has an infinite-dimensional closed subspace whose nonzero vectors are supercyclic.

How to cite

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Salas, Héctor. "Supercyclicity and weighted shifts." Studia Mathematica 135.1 (1999): 55-74. <http://eudml.org/doc/216643>.

@article{Salas1999,
abstract = {An operator (linear and continuous) in a Fréchet space is hypercyclic if there exists a vector whose orbit under the operator is dense. If the scalar multiples of the elements in the orbit are dense, the operator is supercyclic. We give, for Fréchet space operators, a Supercyclicity Criterion reminiscent of the Hypercyclicity Criterion. We characterize the supercyclic bilateral weighted shifts in terms of their weight sequences. As a consequence, we show that a bilateral weighted shift is supercyclic if and only if it satisfies the Supercyclicity Criterion. We exhibit two supercyclic, irreducible Hilbert space operators which are C*-isomorphic, but one is hypercyclic and the other is not. We prove that a Banach space operator which satisfies a version of the Supercyclicity Criterion, and has zero in its left essential spectrum, has an infinite-dimensional closed subspace whose nonzero vectors are supercyclic.},
author = {Salas, Héctor},
journal = {Studia Mathematica},
keywords = {hypercyclic and supercyclic vectors; Hypercyclicity Criterion; Supercyclicity Criterion; weighted shifts; semi-Fredholm operators in Banach spaces; left essential spectrum; basic sequences.; cyclicity; hypercyclicity; supercyclicity; Fréchet spaces; supercyclic bilateral shifts; weight sequences; -isomorphisms},
language = {eng},
number = {1},
pages = {55-74},
title = {Supercyclicity and weighted shifts},
url = {http://eudml.org/doc/216643},
volume = {135},
year = {1999},
}

TY - JOUR
AU - Salas, Héctor
TI - Supercyclicity and weighted shifts
JO - Studia Mathematica
PY - 1999
VL - 135
IS - 1
SP - 55
EP - 74
AB - An operator (linear and continuous) in a Fréchet space is hypercyclic if there exists a vector whose orbit under the operator is dense. If the scalar multiples of the elements in the orbit are dense, the operator is supercyclic. We give, for Fréchet space operators, a Supercyclicity Criterion reminiscent of the Hypercyclicity Criterion. We characterize the supercyclic bilateral weighted shifts in terms of their weight sequences. As a consequence, we show that a bilateral weighted shift is supercyclic if and only if it satisfies the Supercyclicity Criterion. We exhibit two supercyclic, irreducible Hilbert space operators which are C*-isomorphic, but one is hypercyclic and the other is not. We prove that a Banach space operator which satisfies a version of the Supercyclicity Criterion, and has zero in its left essential spectrum, has an infinite-dimensional closed subspace whose nonzero vectors are supercyclic.
LA - eng
KW - hypercyclic and supercyclic vectors; Hypercyclicity Criterion; Supercyclicity Criterion; weighted shifts; semi-Fredholm operators in Banach spaces; left essential spectrum; basic sequences.; cyclicity; hypercyclicity; supercyclicity; Fréchet spaces; supercyclic bilateral shifts; weight sequences; -isomorphisms
UR - http://eudml.org/doc/216643
ER -

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