Abstract
For each automorphic composition operator C
ϕ acting on either the Hardy or Bergman Hilbert space of the unit ball in , we show that
is a Toeplitz operator, that
is the inverse of a Toeplitz operator, and that the selfcommutator
is essentially a Toeplitz operator. We then extract spectral and norm information from these identifications, focusing on selfcommutators. For example, in the setting of the Hardy space of the unit disk, we obtain complete descriptions of the spectrum, essential spectrum, and point spectrum for selfcommutators of automorphic composition operators, which reveal that the spectrum and essential spectrum coincide.