Unitary part of a contraction

Abstract

For a contraction A on a Hilbert space H, we define the index j(A) (resp.. k(A)) as the smallest nonnegative integer j (resp., k) such that ker(I - A(j)*A(j)) (resp., ker(I - A(k)*A(k)) boolean AND ker(I - A(k)*A(k)*)) equals the subspace of H on which the unitary part of A acts. We show that if n = dim H < infinity, then j(A) <= n (resp., k(A) <= left particularn/2left particular). and the equality holds if and only if A is of class S, (resp., one of the three conditions is true: (1) A is of class S, (2) n is even and A is completely nonunitary with parallel to A(n-2)parallel to = 1 and parallel to A(n-1)parallel to < 1. and (3) n is even and A = U circle times A', where U is unitary on a one-dimensional space and A' is of class S(n-1)). (C) 2010 Elsevier Inc. All rights reserved.