Equality of numerical ranges of matrix powers

Abstract

For an n-by-n matrix A, we determine when the numerical ranges W(A(k)), k >= 1, of powers of A are all equal to each other. More precisely, we show that this is the case if and only if A is unitarily similar to a direct sum B circle plus C, where B is idempotent and C satisfies W(C-k) subset of W (B) for all k >= 1. We then consider, for each n >= 1, the smallest integer k(n) for which every n-by-n matrix A with W(A) = W(A(k)) for all k, 1 <= k <= k(n), has an idempotent direct summand. For each n >= 1, let p(n) be the largest prime less than or equal to n + 1. We show that (1) k(n) >= p(n) for all n, (2) if A is normal of size n, then W(A) = W(A(k)) for all k, 1 <= k <= p(n), implies A having an idempotent summand, and (3) k(1) = 2 and k(2) = k(3) = 3. These lead us to ask whether k(n) = p(n) holds for all n >= 1. (C) 2019 Elsevier Inc. All rights reserved.