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The Essential Norm of a Composition Operator

Joel H. Shapiro
Annals of Mathematics
Second Series, Vol. 125, No. 2 (Mar., 1987), pp. 375-404
Published by: Annals of Mathematics
DOI: 10.2307/1971314
Stable URL: http://www.jstor.org/stable/1971314
Page Count: 30
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The Essential Norm of a Composition Operator
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Abstract

We express the essential norm of a composition operator on the Hardy space H2 as the asymptotic upper bound of a quantity involving the Nevanlinna counting function of the inducing map. There results a complete function theoretic characterization of the compact composition operators on H2. Similar results hold for the weighted Bergman spaces of the unit disc. As consequences we obtain: (i) estimates of the essential norm of a composition operator in terms of the angular derivative of its inducing map; (ii) a new proof of a recently obtained characterization of the compact composition operators on the weighted Bergman spaces; and (iii) a new proof of a peak set theorem for holomorphic Lipschitz functions.

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