Finite Blaschke products of contractions

Abstract

Let A be a contraction on Hilbert space H and phi a finite Blaschke product. In this paper, we consider the problem when the norm of phi(A) is equal to 1. We show that (1) parallel tophi(A)parallel to = 1 if and only if parallel toA(k)parallel to = 1, where k is the number of zeros of phi counting multiplicity, and (2) if H is finite-dimensional and A has no eigenvalue of modulus 1, then the largest integer I for which parallel toA(l)parallel to = 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. (C) 2003 Elsevier Science Inc. All rights reserved.