Hardy spaces that support no compact composition operators

Abstract. We consider, for G a simply connected domain and 0 < p < 1, the Hardy space Hp(G) formed by flxing a Riemann map ¿ of the unit disc onto G, and demanding of functions F holomorphic on G that the integrals of jF jp over the curves ¿(fjzj = rg) 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 re°ected in our Main Theorem: Hp(G) supports compact composition operators if and only if @G has flnite one-dimensional Hausdorfi measure. Our work is inspired by an earlier result of Matache [14], 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 \@G has flnite Hausdorfi 1-measure " replaced by \G has flnite area. " Finally, we characterize those domains G for which every composition operator on either the Hardy or the Bergman spaces is bounded. 1