Hardy Spaces That Support No Compact Composition Operators

We consider, for G a simply connected domain and 0 < p < ∞, the Hardy space H(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 ∂G has finite one-dimensional Hausdorff measure. Our work is inspired by an earlier result of Matache [14], who showed that the H 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 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.