What do composition operators know about inner functions?

Joel H. Shapiro

Monatshefte für Mathematik 130 (2000), 57--70.


Abstract: This 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 function.

When the inducing map does not fix the origin there is an interesting surprise!


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