Commutativity of Toeplitz operators on the harmonic Dirichlet space (Q2465872)

Let \({\mathcal D}\) be the Dirichlet space and \({\mathcal D}_h={\mathcal D}+\overline{\mathcal D}\) be the harmonic Dirichlet space on the unit disk \({\mathbb D}\). A nonnegative measure \(\mu\) on \({\mathbb D}\) is called a \({\mathcal D}\)-Carleson measure if \(\int_{\mathbb D}| f|^2\,d\mu\leq C\| f\|^2\) for all \(f\in{\mathcal D}\) and some universal constant \(C>0\). Denote by \({\mathcal M}\) the set of all harmonic functions on \({\mathbb D}\) which can be represented in the form \(u=f+\overline{g}\) with \(f,g\in H^\infty({\mathbb D})\) such that \(| f'|^2\,dA\) and \(| g'|^2\,dA\) are \({\mathcal D}\)-Carleson measures. Here, \(dA\) denotes the normalized area measure on \({\mathbb D}\). Let \(T_u\) denote the Toeplitz operator with a symbol \(u\in{\mathcal M}\) on \({\mathcal D}_h\). Suppose that \(u,v\in{\mathcal M}\). It is proved that \(T_uT_v=T_vT_u\) if and only if a nontrivial linear combination of \(u\) and \(v\) is constant on \({\mathbb D}\). Further, it is shown that \(T_uT_v=T_{uv}\) if and only if either \(u\) or \(v\) is constant.