Self-commutators of invertible weighted composition operators on H²

In this paper, we consider the self-commutator of an invertible weighted composition operator W_{f, φ} on the Hardy space H² where f is continuous on D̄. We show that both the self-commutator [W_f,φ*, W_f,φ] and the anti-self-commutator {W_f,φ*, W_f,φ} are expressed as compact perturbations of Toeplitz operators. Moreover, we give an alternative proof for the result in [2] that W_f,φ is unitary exactly when φ is an automorphism of D and (Formula presented.) where p = φ^-1(0), K_p is the reproducing kernel at p for H², and c is a constant with {pipe}c{pipe} = 1. We next show that when {pipe}f(z){pipe} ≤ {pipe}f(φ(z)){pipe} for all z ∈ D, the weighted composition operator W_f,φ is normal if and only if the composition operator C_φ is unitary and f is constant on D. We also provide some spectral properties of W_f,φ* W_f,φ and W_f,φ W_f,φ*.