Monatshefte für Mathematik

, Volume 130, Issue 1, pp 57–70

What Do Composition Operators Know About Inner Functions?

  • Joel H. Shapiro

DOI: 10.1007/s006050050087

Cite this article as:
Shapiro, J. Mh Math (2000) 130: 57. doi:10.1007/s006050050087


 This paper gives several different ways in which operator norms characterize those composition operators \(\) that arise from holomorphic self-maps ϕ of the unit disc that are inner functions. The setting is the Hardy space H2 of the disc, and the key result is a characterization of inner functions in terms of the asymptotic behavior of the Nevanlinna counting function. The case \(\) offers an interesting surprise.

1991 Mathematics Subject Classification: 47B3330D55
Key words: Composition operatorinner functionessential norm

Copyright information

© Springer-Verlag Wien 2000

Authors and Affiliations

  • Joel H. Shapiro
    • 1
  1. 1.Michigan State University, East Lansing, MI, USAUS