Commuting dual Toeplitz operators on the harmonic Bergman space
Let \(D\) be the unit disk, and \(dA\) be the normalized area measure on \(D\). The harmonic Bergman space \(L^2_h\) is the closed subspace of \(L^2(D, dA)\) which consists of the harmonic functions on \(D\), and let \(Q\) be the orthogonal projection of \(L^2(D, dA)\) onto \(L^2_h\). The authors define the dual Toeplitz operator \(S_{\varphi}\) with symbol \(\varphi \in L^{\infty}(D)\), acting on \((L^2_h)^{\perp}\), by \(S_{\varphi}(f) = (I-Q)(\varphi f)\). The main result of the paper states: Let \(\varphi\), \(\psi\) be bounded harmonic functions on \(D\), then \(S_{\varphi}S_{\psi} = S_{\psi}S_{\varphi}\) if and only if one of the following conditions hold: (1) Both \(\varphi\) and \(\psi\) are analytic on \(D\). (2) Both \(\varphi\) and \(\psi\) are anti-analytic on \(D\). (3) There exist complex constants \(\alpha\) and \(\beta\), both not 0, such that \(\varphi = \alpha \psi + \beta\). A necessary and sufficient condition for \(S_{\varphi}S_{\psi} = S_{\varphi\psi}\) is also given.
scientific article; zbMATH DE number 6499353