Regular orbits and positive directions

Abstract

Let A be a bounded linear operator defined on a separable Banach space X. Then A is said to be supercyclic if there exists a vector x ∈ X (later called supercyclic for A), such that the projective orbit $\{\lambda A^{n} x\,:\,n \in {\mathbb{N}},\,\lambda \in {\mathbb{C}}\}$ is dense in X. On the other hand, A is said to be positive supercyclic if for each supercyclic vector x, the positive projective orbit, $\{rA^nx\,:\, r \in {\mathbb{R}}_{+},\,n \in {\mathbb{N}}\}$ is dense in X. Sometimes supercyclicity and positive supercyclicity are equivalent. The study of this relationship was initiated in [14] by F. León and V. Müller. In this paper we study positive supercyclicity for operators A of the form $A=T \oplus \alpha 1_{{\mathbb{C}}}$, with $\alpha \in {\mathbb{C}}{\setminus}\{0\}$, defined on $X \oplus {\mathbb{C}}$. We will see that such a problem is related with the study of regular orbits. The notion of positive directions will be central throughout the paper.