## Mathematics and Statistics Faculty Publications

#### Title

Cyclicity of Vectors with Orbital Limit Points for Backward Shifts

#### Document Type

Article

#### Abstract

On a separable, infinite dimensional Banach space X, a bounded linear operator T : X → X is said to be hypercyclic, if there exists a vector x in X such that its orbit Orb(T, x) = {x, Tx, T2x, …} is dense in X. In a recent paper (Chan and Seceleanu in J Oper Theory 67:257–277, 2012), it was shown that if a unilateral weighted backward shift has an orbit with a single non-zero limit point, then it possesses a dense orbit, and hence the shift is hypercyclic. However, the orbit with the non-zero limit point may not be dense, and so the vector x inducing the orbit need not be hypercyclic. Motivated by this result, we provide conditions for x to be a cyclic vector.

#### Repository Citation

Chan, Kit C. and Seceleanu, Irina, "Cyclicity of Vectors with Orbital Limit Points for Backward Shifts" (2014). *Mathematics and Statistics Faculty Publications*. 32.

https://scholarworks.bgsu.edu/math_stat_pub/32

#### Publication Date

2014

#### Publication Title

Integral Equations and Operator Theory