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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.

Keywords:  Orbital limit points, hypercyclicity, shift operators

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