Matrix powers with circular numerical range

Abstract

Let K-2 = [GRAPHICS}, K-n be the n x n weighted shift matrix with weights root 2, [GRAPHICS}, root 2 for all n >= 3, and K-infinity be the weighted shift operator with weights root 2, 1, 1, 1, .... In this paper, we show that if an n x 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 is an element of C : vertical bar z vertical bar <= 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 ->infinity) parallel to T(n)x parallel to = root 2 for some unit vector x is an element of H if and only if T is unitarily similar to an operator of the form K-infinity circle plus T' with w(T') <= 1. (C) 2020 Elsevier Inc. All rights reserved.