Matrix powers with circular numerical range

Hwa Long Gau, Kuo-Zhong Wang*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Let [Formula presented], Kn 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(Ak)=W(A) for all 1≤k≤n, then W(A) cannot be a (nondegenerate) circular disc. Moreover, we also show that W(A)=W(An−1)={z∈C:|z|≤1} if and only if A is unitarily similar to Kn. Finally, we prove that if T is a numerical contraction on an infinite-dimensional Hilbert space H, then limn→∞⁡‖Tnx‖=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 languageEnglish
Pages (from-to)190-211
Number of pages22
JournalLinear Algebra and Its Applications
Volume603
DOIs
StatePublished - 15 Oct 2020

Keywords

  • Numerical contraction
  • Numerical radius
  • Numerical range

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