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Hypercyclicity and Supercyclicity for Invertible Bilateral Weighted Shifts

To appear in Proc. Amer. Math. Soc.

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A bounded linear operator T is hypercyclic if there is a vector with dense orbit; so if T acts on a Hilbert space H, then T is hypercyclic if there is a vector x in H such that {x, Tx, T^2x, ...} is dense in H.  An operator is supercyclic if there is a one-dimensional subspace with dense orbit; thus there exists a vector x such that {cT^nx : n > 0 and c is a complex scalar} is dense.

Hector Salas has characterized the Bilateral weighted shifts that are hypercyclic and those that are supercyclic in terms of their weight sequence.  However the characterization is (necessarily) rather complicated, involving several quantifiers.

In this paper we give a rather simple, natural condition characterizing the hypercyclicity and supercyclicity for invertible weighted shifts.