Integral Equations and Operator Theory

, Volume 78, Issue 2, pp 225–232 | Cite as

Cyclicity of Vectors with Orbital Limit Points for Backward Shifts

Article

Abstract

On a separable, infinite dimensional Banach space X, a bounded linear operator T : XX 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.

Mathematics Subject Classification (2010)

Primary 47A16 Secondary 47B37 

Keywords

Backward weighted shifts hypercyclicity orbital limit points cyclic vector 

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsBowling Green State UniversityBowling GreenUSA
  2. 2.Department of MathematicsBridgewater State UniversityBridgewaterUSA

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