#### Title

Cyclicity of Vectors with Orbital Limit Points for Backward Shifts

#### Publication Date

2014

#### 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*, *T*^{2}*x*, …} 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.

#### Original Citation

Chan, K. & Seceleanu, I. (2014). Cyclicity of Vectors with Orbital Limit Points for Backward Shifts. *I**ntegral Equations and Operator Theory,* 78(2), 225-232. https://doi.org/10.1007/s00020-013-2100-2

#### Identifier

#### Virtual Commons Citation

Chan, Kit and Seceleanu, Irina (2014). Cyclicity of Vectors with Orbital Limit Points for Backward Shifts. In *Mathematics Faculty Publications.* Paper 38.

Available at: https://vc.bridgew.edu/math_fac/38