Orbits of Cesaro type operators

Abstract

A bounded linear operator T on a Banach space X is called hypercyclic if there exists a vector x is an element of X such that its orbit, {T(n)x}, is dense in X. In this paper we show hypercyclic properties of the orbits of the Cesaro operator defined on different spaces. For instance, we show that the Cesaro operator defined on L(p)[0, 1] (1 < p < infinity) is hypercyclic. Moreover, it is chaotic and it has supercyclic subspaces. On the other hand, the Cesaro operator defined on other spaces of functions behave differently. Motivated by this, we study weighted Cesaro operators and different degrees of hypercyclicity are obtained. The proofs are based on the classical Muntz-Szasz theorem. We also propose problems and give new directions.