## Abstract

Let [Formula presented], K_{n} be the n×n weighted shift matrix with weights 2,1,…,1︸n−3,2 for all n≥3, and K_{∞} be the weighted shift operator with weights 2,1,1,1,…. In this paper, we show that if an n×n nonzero matrix A satisfies W(A^{k})=W(A) for all 1≤k≤n, then W(A) cannot be a (nondegenerate) circular disc. Moreover, we also show that W(A)=W(A^{n−1})={z∈C:|z|≤1} if and only if A is unitarily similar to K_{n}. Finally, we prove that if T is a numerical contraction on an infinite-dimensional Hilbert space H, then lim_{n→∞}‖T^{n}x‖=2 for some unit vector x∈H if and only if T is unitarily similar to an operator of the form K_{∞}⊕T^{′} with w(T^{′})≤1.

Original language | English |
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Pages (from-to) | 190-211 |

Number of pages | 22 |

Journal | Linear Algebra and Its Applications |

Volume | 603 |

DOIs | |

State | Published - 15 Oct 2020 |

## Keywords

- Numerical contraction
- Numerical radius
- Numerical range