Finite Blaschke products of contractions

Let A be a contraction on Hilbert space H and φ a finite Blaschke product. In this paper, we consider the problem when the norm of φ(A) is equal to 1. We show that (1) ∥φ(A)∥=1 if and only if ∥A ^k∥=1, where k is the number of zeros of φ counting multiplicity, and (2) if H is finite-dimensional and A has no eigenvalue of modulus 1, then the largest integer l for which ∥A ^l∥=1 is at least m/(n-m), where n=dim H and m=dim ker(I-A*A), and, moreover, l=n-1 if and only if m=n-1.