Computational Methods and Function Theory 6 (2006), No. 1, 37--50
Copyright Heldermann Verlag 2006
Christopher Hammond
cnham@conncoll.edu , Department of Mathematics and Computer Science, Connecticut College, Box 5384, 270 Mohegan Avenue, New London, CT 06320, U.S.A.
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||. Furthermore, we will use our knowledge of composition operators to establish an apparently new result pertaining to the zeros of hypergeometric series.
Keywords: Composition operator, operator norm, hypergeometric series.
MSC 2000: Primary 47B33; Secondary 33C05.
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