Matrix powers with circular numerical range

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(Aⁿ⁻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→∞⁡‖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.