Finite Blaschke products of contractions

✍ Scribed by Hwa-Long Gau; Pei Yuan Wu

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 128 KB
• Volume: 368
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(02)00697-3

✦ Synopsis

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/(nm), where n = dim H and m = dim ker(I -A * A), and, moreover, l = n -1 if and only if m = n -1.