# Supercyclicity and weighted shifts

Studia Mathematica (1999)

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

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topSalas, 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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