Algebraic properties of dual Toeplitz operators on the orthogonal complement of the Dirichlet space
The Dirichlet space \({\mathcal D}\) is the space of all analytic functions in the unit disk that belong to the Sobolev space \(W^{1,2}\). Let \(Q\) denote the projection of \(W^{1,2}\) onto \({\mathcal D}^\perp\). For a function \(\varphi\) in the Sobolev space \(W^{1,\infty}\), let \(S_\varphi\) denote the dual Toeplitz operator defined by \(S_\varphi(u)=Q(\varphi u)\) for \(u\in{\mathcal D}^\perp\). In the paper under review, the following four theorems are proved. Theorem 1. Suppose that \(\varphi,\psi\in W^{1,\infty}\) are harmonic functions. Then \(S_\varphi S_\psi=S_\psi S_\varphi\) if and only if one of the following conditions holds: (i) both \(\varphi\) and \(\psi\) are analytic; (ii) a nontrivial linear combination of \(\varphi\) and \(\psi\) is analytic. Theorem 2. Suppose that \(\varphi\) is a nonconstant bounded analytic function and \(\psi\in W^{1,\infty}\). If \(S_\varphi\) and \(S_\psi\) commute, then \(\psi\) is analytic. Theorem 3. Suppose that \(\varphi,\psi\in W^{1,\infty}\) are harmonic functions. Then \(S_\varphi S_\psi=S_{\varphi\psi}\) if and only if one of the following statements holds: (i) \(\varphi\) is analytic; (ii) \(\psi\) is constant. Theorem 4. Suppose that \(\varphi,\psi\in W^{1,\infty}\) and \(S_\varphi S_\psi\) is a dual Toeplitz operator \(S_u\) for some \(u\in W^{1,\infty}\). Then \(\varphi\psi=u\).
scientific article; zbMATH DE number 5485330