Commutators of weighted composition operators

In this paper, we prove that if the composition symbols φ and ψ are linear fractional non-automorphisms of such that φ(ζ) and ψ(ζ) belong to ∂ for some ζ ∈ ∂ and u, v ∈ H^∞ are continuous on ∂ with u(ζ)v(ζ) ≠ 0, then [{W^*v, psi, Wu,φ] is compact on H² if and only if ζ is the common boundary fixed point of φ and ψ and one of the following statements holds: (i) both φ and ψ are parabolic; (ii) both φ and ψ are hyperbolic and another fixed point of φ is 1=w where w is the fixed point of ψ other than ζ. We also study the commutant of a weighted composition operator on H². We verify that if φ is an analytic self-map of with Denjoy-Wolff point b ∈ and u ∈ H^∞\{0}, then every weighted composition operator in the commutant {Wu, φ}′ has {f ∈ H² : f(b) = 0} as its nontrivial invariant subspace.