Abstract: We
consider, for G a simply connected domain and p finite, the
Hardy space H^{p}(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:
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 halfplanes 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 Hardyspace
condition ``boundary of G has finite Hausdorff 1measure'' 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.
