What do composition operators
know about inner functions?
Joel H. Shapiro
Monatshefte für Mathematik
130 (2000), 57--70.
paper gives several different ways in which operator norms characterize
those composition operators that arise from holomorphic self-maps
of the unit disc that are inner functions.
The setting is the Hardy space H^2 of the
disc, and the key result is a characterization of inner functions
in terms of the asymptotic behavior of the Nevanlinna counting
When the inducing map does not fix the origin
there is an interesting surprise!
Download .pdf file (192K):