Supercyclicity and weighted shifts

Héctor N. Salas

Studia Mathematica, Tome 135 (1999) no. 1, pp. 55-74

Résumé

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.

Détail

DOI : 10.4064/sm-135-1-55-74

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