Commuting dual Toeplitz operators on the orthogonal complement of the Dirichlet space
Let \(D\) be the open unit disk in the complex plane and let \(dA\) denote the normalized Lebesgue measure. The Sobolev space \(W^{1,2}\) consists of functions \(u:D\to\mathbb{C}\) with the weak partial derivatives of order 1 and the norm \[ \|u\|_{\frac{1}{2}}=\left(\left|\int_D u dA\right|^2+\int_D\left(\left|\frac{\partial u}{\partial z}\right|^2+\left|\frac{\partial u}{\partial\overline{z}}\right|^2\right)dA\right)^{\frac{1}{2}}. \] The Dirichlet space \({\mathcal D}_0\) is the closed subspace of the Hilbert space \(W^{1,2}\) that consists of all holomorphic functions \(f\in W^{1,2}\) with \(f(0)=0\). Let \[ W^{1,\infty}(D)=\left\{\varphi\in W^{1,2}: \varphi,\;\frac{\partial u}{\partial z},\;\frac{\partial u}{\partial\overline{z}}\in L^\infty(D)\right\}. \] Given \(\varphi\in W^{1,\infty}(D)\), the dual Toeplitz operator \(S_\varphi:{\mathcal D}_0^\bot\to{\mathcal D}_0^\bot\) is defined by \(S_\varphi u=Q(\varphi u)\) for \(u\in{\mathcal D}_0^\bot\), where \(Q\) is the orthogonal projection from \(W^{1,2}\) onto \({\mathcal D}_0^\bot\), the orthogonal complement of \({\mathcal D}_0\) in \(W^{1,2}\). Theorem 1: Suppose that \(\varphi,\psi\in W^{1,\infty}(D)\) are harmonic functions. Then \(S_\varphi\) and \(S_\psi\) commute if and only if either both \(\varphi\) and \(\psi\) are holomorphic on \(D\), or both \(\varphi\) and \(\psi\) are antiholomorphic on \(D\), or a nontrivial linear combination of \(\varphi\) and \(\psi\) is constant on \(D\). Theorem 2: Suppose that \(\varphi,\psi\in W^{1,\infty}(D)\) are harmonic functions. Then \(S_\varphi S_\psi=S_{\varphi\psi}\) if and only if either \(\varphi\) is holomorphic on \(D\), or \(\psi\) is antiholomorphic on \(D\). Theorem 3: If \(\varphi,\psi\in W^{1,\infty}(D)\), and \(S_\varphi S_\psi\) is a finite rank perturbation of a dual Toeplitz operator \(S_u\) for some \(u\in W^{1,\infty}(D)\), then \(\varphi\psi=u\).
scientific article; zbMATH DE number 5629497