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ArticleOriginal scientific text

Title

Supercyclicity and weighted shifts

Authors 1

Affiliations

  1. Departamento de Matemáticas, Universidad de Puerto Rico, Mayagüez, Puerto Rico 00681

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.

Keywords

ENG
hypercyclic and supercyclic vectors, Hypercyclicity Criterion, Supercyclicity Criterion, weighted shifts, semi-Fredholm operators in Banach spaces, left essential spectrum, basic sequences.

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Studia Mathematica
1999/135/1
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Pages:
55-74
Main language of publication
English
Received
1998-04-21
Accepted
1998-10-23
Published
1999
Exact and natural sciences