Constant norms and numerical radii of matrix powers

For an n-by-n complex matrix A, we consider conditions on A for which the operator norms ||A^k|| (resp., numerical radii w(A^k)), k ≥ 1, of powers of A are constant. Among other results, we show that the existence of a unit vector x in Cⁿ satisfying |〈A^k x,x〉| = w(A^k)=w(A) for 1 ≤ k ≤ 4 is equivalent to the unitary similarity of A to a direct sum (Formula Presented), where |λ| = 1, B is idempotent, and C satisfies w(C^k) ≤ w(B) for 1 ≤ k ≤ 4. This is no longer the case for the norm: there is a 3-by-3 matrix A with ||A^k x|| = ||A^k|| =√2 for some unit vector x and for all k ≥ 1, but without any nontrivial direct summand. Nor is it true for constant numerical radii without a common attaining vector. If A is invertible, then the constancy of ||A^k || (resp., w(A^k)) for k = ±1,±2,… is equivalent to A being unitary. This is not true for invertible operators on an infinite-dimensional Hilbert space.