Orbits of Cesàro type operators

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, {T n x}, 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 onMoreover, 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.

On the continuous Cesàro operator in certain function spaces

“…Recall that a bounded linear operator T , dened on a separable Banach space X (or, more gen-1 < p < ∞, is hypercyclic and chaotic, [23], and that it is not (weakly) supercyclic in L 2 (R + ), [17]. On the other hand, C is not supercyclic (hence, not hypercyclic) on C([0, 1]), [23].…”

Positivity in the theory of supercyclic operators

“…In Section 3 it is proved that the classical Cesàro operator is hypercyclic on L p [0, 1], 1 < p < ∞, nevertheless (thanks to the Positive Supercyclicity Theorem), this result is not true on C[0, 1] with the uniform norm. This result is extended in [29], where the authors extensively study the hypercyclic and supercyclic properties of integral operators. Related results on supercyclicity of integral operators appeared in [17].…”

“…The results in [29] suggest that if the kernel K is regular then the orbit of V K is also regular. However, if the behaviour of K is irregular then the orbit of some vector under V n K can be chaotic.…”

“…However, if the behaviour of K is irregular then the orbit of some vector under V n K can be chaotic. Nevertheless, for different kernels it is possible to obtain different grades of hypercyclicity (see [29]). …”

“…Finally, let us observe that the functions in A correspond to the eigenvalues Nevertheless, this result is not true on C[0, 1] with the supremum norm (see [29]). Moreover, the Cesàro operator C is not supercyclic on C[0, 1].…”