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Abstract: We consider, for G a simply connected domain and p finite , the Hardy space H^p(G) formed by fixing a Riemann map t of the unit disc onto G, and demanding of functions F holomorphic on G that the integrals of |F|^p over the curves t({|z|=r}) be bounded for 0<r<1. The resulting space is usually not the one obtained from the classical Hardy space of the unit disc by conformal mapping. This is reflected in our Main Theorem: H^p(G) supports compact composition operators if and only if the boundary of G has finite one-dimensional Hausdorff measure. Our work is inspired by an earlier result of Matache who showed that the H^p spaces of half-planes support no compact composition operators. Our methods provide a lower bound for the essential spectral radius which shows that the same result holds with ``compact'' replaced by ``Riesz''. We prove similar results for Bergman spaces, with the Hardy-space condition ``boundary of G has finite Hausdorff 1-measure'' replaced by ``G has finite area.'' Finally, we characterize those domains G for which every composition operator on either the Hardy or the Bergman spaces is bounded. |
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