Journal of Operator Theory

Volume 67, Issue 1, Winter 2012 pp. 257-277.

Hypercyclicity of shifts as a zero-one law of orbital limit points

Authors: Kit Chan (1) and Irina Seceleanu (2)
Author institution: (1) Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, 43403, U.S.A.
(2) Department of Mathematics and Computer Science, Bridgewater State University, Bridgewater, 02325, U.S.A.

Summary: On a separable, infinite dimensional Banach space

$X$ , a bounded linear operator

$T:X \rightarrow X$ is said to be \textit{hypercyclic} if there exists a vector

$x$ in

$X$ such that its orbit

$\mathrm{Orb}(T,x)=\{x, Tx, T^2x, \ldots\}$ is dense in

$X$ . However, for a unilateral weighted backward shift or a bilateral weighted shift

$T$ to be hypercyclic, we show that it suffices to merely require the operator to have an orbit

$\mathrm{Orb}(T,x)$ with a non-zero limit point.

Contents Full-Text PDF