Abstract
A bounded linear operator T on a Banach space X is called hypercyclic if there exists a vector x ∈ X such that its orbit, {Tnx }, is dense in X. In this paper we show hypercyclic properties of the orbits of the Cesàro operator defined on different spaces. For instance, we show that the Cesàro operator defined on Lp [0, 1] (1 < p < ∞) is hypercyclic. Moreover, it is chaotic and it has supercyclic subspaces. On the other hand, the Cesàro operator defined on other spaces of functions behave differently. Motivated by this, we study weighted Cesàro operators and different degrees of hypercyclicity are obtained. The proofs are based on the classical Müntz–Szász theorem. We also propose problems and give new directions (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
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