Defect indices of powers of a contraction

Abstract

Let A be a contraction on a Hilbert space H. The defect index d(A) of A is, by definition, the dimension of the closure of the range of l - A*A. We prove that (1) d(An) <= nd(A) for all n >= 0, (2) if, in addition, A(n) converges to 0 in the strong operator topology and d(A) = 1, then d(An) = n for all finite n, 0 <= n <= dim H, and (3) d(A) = d(A)* implies d(An) = d(An)* for all n >= 0. The norm-one index k(A) of A is defined as sup{n >= 0 : parallel to A(n)parallel to = 1}. When dim H = m < infinity, a lower bound for k(A) was obtained before: k(A) >= (m/d(A)) - 1. We show that the equality holds if and only if either A is unitary or the eigenvalues of A are all in the open unit disc, d(A) divides m and d(An) = nd(A) for all n, 1 <= n <= m/d(A). We also consider the defect index of f(A) for a finite Blaschke product f and show that d(f(A)) = d(An), where n is the number of zeros off. (C) 2009 Elsevier Inc. All rights reserved.