Hardy spaces that support no compact composition operators


 by

Joel H. Shapiro and Wayne Smith

 

Journal of Functional Analysis 205 (2003) 62--89


 Abstract: We consider, for G a simply connected domain and p finite, the Hardy space Hp(G) formed by fixing a Riemann map τ of the unit disc onto G, and demanding of functions F holomorphic on G that the integrals of |F|p over the curves τ({|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:

Hp(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 Hp 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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