Hausdorff measures, capacities and compact composition operators |
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Authors: | Eva A Gallardo-Gutiérrez María J González |
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Institution: | (1) Departamento de Matemáticas, Universidad de Zaragoza, Plaza San Francisco s/n, 50009 Zaragoza, Spain;(2) Departamento de Matemáticas, Universidad de Cádiz, Apartado 40, 11510 Puerto Real (Cádiz), Spain |
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Abstract: | It is shown that there exist analytic self-maps ϕ of the unit disc inducing compact composition operators on the Hardy space , 1 ≤ p < ∞ such that the Hausdorff dimension of the set is one; sharpening a classical result due to Schwartz. Moreover, the same holds in the weighted Dirichlet spaces with 0 < α < 1. As a consequence, we deduce that there exist symbols ϕ inducing compact composition operators on such that the α-capacity of Eϕ is positive, which is no longer true for those just inducing Hilbert-Schmidt composition operators on .
First author is partially supported by Plan Nacional I+D grant no. BFM2003-00034, and Gobierno de Aragón research group Análisis Matemático y Aplicaciones, ref. DGA E-64 . Second author is partially supported by Plan Nacional I+D grant no. BFM2002-00571 and Junta de Andalucía
RNM-314. |
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Keywords: | Composition operator Weighted Dirichlet spaces Hausdorff measures |
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