Selfcommutators of automorphic composition operators

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.

Keywords:

• Composition operator
• Commutator
• Essentially Toeplitz
• Essentially normal

Keywords:

• 2000 Mathematics Subject Classifications: Primary 47B33
• Secondary 47B35
• 47B20
• 47B47