Commutators of weighted composition operators (Q2874679)

For an analytic self-map \(\varphi\) of the unit disk \(D\) and any analytic function \(u\) on \(D\), define an operator \(W_{u,\varphi}\) by \(W_{u,\varphi}=uf\circ\varphi\). These are usually called weighted composition operators. The main result of the paper states that, if the composition symbols \(\varphi\) and \(\psi\) are linear fractional non-automorphisms of \(D\) such that \(\varphi(\zeta)\) and \(\psi(\zeta)\) belong to \(\partial D\) for some \(\zeta\in\partial D\), and if \(u\) and \(v\) are functions in the disk algebra such that \(u(\zeta)v(\zeta)\not=0\), then the commutator \([W^*_{v,\psi},W_{u,\varphi}]\) is compact on the Hardy space \(H^2\) if and only if \(\zeta\) is the common boundary fixed point of \(\varphi\) and \(\psi\) and one of the following conditions holds: (i) both \(\varphi\) and \(\psi\) are parabolic, (ii) both \(\varphi\) and \(\psi\) are hyperbolic and another fixed point of \(\varphi\) is \(1/\bar w\), where \(w\) is the fixed point of \(\psi\) other than \(\zeta\). The paper also shows that, if \(b\) is a Denjoy-Wolff point of \(\varphi\) and \(u\in H^\infty\) is not identically zero, then every weighted composition operator in the commutant of \(W_{u,\varphi}\) has \(\{f\in H^2:f(b)=0\}\) as an invariant subspace.