On the selfcommutator of automorphic composition operators

Our principal interest here is in the selfcommutator and the anti selfcommutator of an automorphic composition operator C-phi which denoted by [C-phi*; C-phi] and {C-phi*; C-phi}, and defined by [C-phi*; C-phi] = C-phi*C-phi - C phi C phi* and {C-phi*; C-phi} = C-phi*C-phi + C phi C phi* respectively. In this work, we compute an explicit expression for the adjoint of a composition operator on H-2 induced by a conformal automorphism and the starting point for our analysis of [C-phi*; C-phi] and {C-phi*; C-phi} is this formula. Also, we compute the spectrum and norm of [C-phi*; C-phi] and {C-phi*; C-phi} for involution automorphism phi.

Automated summary

The focus of this paper is on the selfcommutator and the anti selfcommutator of an automorphic composition operator. The adjoint of a composition operator induced by a conformal automorphism is computed, providing a starting point for the analysis of the selfcommutator and the anti selfcommutator. Additionally, the spectrum and norm of these operators for involution automorphism phi are computed.