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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 infinitedimensional 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 

Pages (fromto)  190211 
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
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Dive into the research topics of 'Matrix powers with circular numerical range'. Together they form a unique fingerprint.Projects
 1 Finished

A Study on Matrices with Circular Numerical Ranges
Gau, H.L. (PI)
1/08/19 → 31/07/20
Project: Research