Journal of Operator Theory

Volume 58, Issue 1, Summer 2007 pp. 39-62.

Non-weakly supercyclic operators

Authors: Alfonso Montes-Rodr\'{i}guez (1) and Stanislav A. Shkarin (2)
Author institution: (1) Departamento de An\'alisis Matem\'atico, Universidad de Sevilla, Sevilla, 41080, Spain
(2) Department of Mathematics, King's College London, London, WC2R 2LS, UK

Summary: Several methods based on an easy geometric argument are provided to prove that a given operator is not weakly supercyclic. The methods apply to different kinds of operators like composition operators or bilateral weighted shifts. In particular, it is shown that the classical Volterra operator is not weakly supercyclic on any of the $L^p[0,1]$ spaces, $1\leqslant p < \infty$. This is in contrast with the fact that the Volterra operator, extended in a natural way to certain Hilbert spaces, is hypercyclic. With the help of Gaussian measures, a general theorem of non-weak supercyclicity is proved, which can be applied to bilateral shifts or analytic functions of the Volterra operator. For instance, it is shown that a weighted bilateral shift acting on $\ell^p({\mathbb Z})$, $1\leqslant p< 2$, is weakly supercyclic if and only if it is supercyclic.

Keywords: Weakly supercyclic operators, cyclic operators, Volterra operator, bilateral weighted shifts, Gaussian measures, composition operators

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