Power r-cyclicity

Abstract EN

In this paper, we prove that any bounded linear operator on a separable Banach space is circle- cyclic if and only if it is hyper cyclic.

As a continuation of studying cyclic phenomena we define and study a new concept called a power r-cyclic operator.

We show that any power r-cyclic operator is super cyclic, but the converse need not be true in general.

We give an example of a power r-cyclic operator which is not hyper cyclic.

Also we give necessary and sufficient conditions for the power r-cyclic operator to be hyper cyclic and we give necessary and sufficient conditions for the operator to be power r-cyclic.

Finally, we get some results concerning some spectral properties of power r-cyclic operators.