Zeros of Hypergeometric Functions and the Norm of a Composition Operator

Abstract

Let φ be an analytic self-map of the unit disk; let Cφ denote the corresponding composition operator acting on the Hardy space H2. Although the precise value of ∥Cφ∥ is quite difficult to calculate, some progress has been made in the case when φ is a linear fractional map. A recent paper by Basor and Retsek demonstrates a connection between the norm of such an operator and the zeros of a particular hypergeometric series. Here we will pursue this line of inquiry further. We shall appeal to several results relating to hypergeometric series — many of which are quite old — to deduce more information about the norm of a composition operator, in particular about the spectrum of Cφ*Cφ. Furthermore, we will use our knowledge of composition operators to establish an apparently new result pertaining to the zeros of hypergeometric series.