### Abstract

Let A be a unilateral (resp., bilateral) weighted shift with weights wn, n≤0 (resp., -∞<n<∞). Eckstein and Rácz showed before that A has its numerical range W(A) contained in the closed unit disc if and only if there is a sequence {a_{n}}_{n=0}^{∞} (resp., {a_{n}}_{n=0}^{∞}) in [-1,1] such that |w_{n}|2=(1-a_{n})(1+an+1) for all n. In terms of such a_{n}'s, we obtain a necessary and sufficient condition for W(A) to be open. If the w_{n}'s are periodic, we show that the a_{n}'s can also be chosen to be periodic. As a result, we give an alternative proof for the openness of W(A) for an A with periodic weights, which was first proven by Stout. More generally, a conjecture of his on the openness of W(A) for A with split periodic weights is also confirmed.

Original language | English |
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Pages (from-to) | 897-909 |

Number of pages | 13 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 381 |

Issue number | 2 |

DOIs | |

State | Published - 15 Sep 2011 |

### Keywords

- Bilateral weighted shift
- Numerical contraction
- Numerical radius
- Numerical range
- Unilateral weighted shift

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## Cite this

*Journal of Mathematical Analysis and Applications*,

*381*(2), 897-909. https://doi.org/10.1016/j.jmaa.2011.04.010