CONSTANT NORMS AND NUMERICAL RADII OF MATRIX POWERS

Abstract

For an n-by-n complex matrix A, we consider conditions on A for which the operator norms parallel to A(k)parallel to (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 C-n satisfying vertical bar < A(k)x,x >vertical bar = w(A(k)) = w(A) for 1 <= k <= 4 is equivalent to the unitary similarity of A to a direct sum lambda B circle plus C, where vertical bar lambda vertical bar = 1, B is ideinpotent, 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 parallel to A(k)x parallel to = parallel to A(k)parallel to = root 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 parallel to A(k)parallel to (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.