Equality of numerical ranges of matrix powers

Abstract

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 ⊕ C , where B is idempotent and C satisfies W ( C k ) ⊆ 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 .