Defect indices of powers of a contraction

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 I - A^* A. We prove that (1) d_An ≤ nd_A for all n ≥ 0, (2) if, in addition, Aⁿ 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 : {norm of matrix} Aⁿ {norm of matrix} = 1}. When dim H = m < ∞, 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 of f.