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 Ak =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 Al =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.