Commutators of weighted composition operators

Abstract

In this paper, we prove that if the composition symbols φ and ψ are linear fractional non-automorphisms of 𝔻 such that φ(ζ) and ψ(ζ) belong to ∂𝔻 for some ζ ∈ ∂𝔻 and u, v ∈ H∞ are continuous on ∂𝔻 with u(ζ)v(ζ) ≠ 0, then [Formula: see text] 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 [Formula: see text] 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 𝔻 with Denjoy–Wolff point b ∈ 𝔻 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.