Mathematics Faculty Publications

Title

Hypercyclicity of Shifts as a Zero-one Law of Orbital Limit Points

2012

Article

Abstract

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.

Original Citation

Chan, K., Seceleanu, I. (2012). Hypercyclicity of Shifts as a Zero-one Law of Orbital Limit Points. Journal of Operator Theory, 67(1), 257-277.

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