Regular orbits and positive directions

Regular orbits and positive directions 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Regular orbits and positive directions

, Volume 13 (4) – Feb 6, 2009
12 pages

/lp/springer_journal/regular-orbits-and-positive-directions-8IU1Kn0eyf
Publisher
Birkhäuser-Verlag
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-008-2295-7
Publisher site
See Article on Publisher Site

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.

Journal

PositivitySpringer Journals

Published: Feb 6, 2009

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