Integral Equations and Operator Theory

, Volume 78, Issue 2, pp 225–232

# 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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### References

1. 1.
Bourdon P.S., Feldman N.S.: Somewhere dense orbits are everywhere dense. Indiana Univ. Math. J. 52, 811–819 (2003)
2. 2.
Chan K.C., Seceleanu I.: Hypercyclicity of shifts as a zero-one law of orbital limit points. J. Oper. Theory 67, 257–277 (2012)
3. 3.
Chan K.C., Seceleanu I.: Orbital limit points and hypercyclicity of operators on analytic function spaces. Math. Proc. R. Ir. Acad. 110, 99–109 (2010)
4. 4.
Salas H.: Hypercyclic weighted shifts. Trans. Am. Math. Soc. 347(3), 993–1004 (1995)
5. 5.
Shields A.: Weighted Shift Operators and Analytic Function Theory. In: Topics in Operator Theory, pp. 149–128. Mathematical Surveys Number 13, American Mathematical Society (1974)Google Scholar