Algebraic properties of Toeplitz operators on the harmonic Dirichlet space (Q628689)

By the harmonic Dirichlet space \({\mathcal D}_h\) over the unit disk \(D\) of the complex plane, the authors mean the closed subspace of \({\mathcal L}^{2,1}\) (=\(L^2\)-Sobolev space of order 1 over \(D\)) consisting of harmonic functions. Given \(\varphi\in {\mathcal L}^{2,1}\), the Toeplitz operator \(T_\varphi:{\mathcal D}_h\to {\mathcal D}_h\) with symbol \(\varphi\) is densely defined by \(T_\varphi f= Q(\varphi f)\) whenever \(f\varphi\in{\mathcal L}^{2,1}\), where \(Q\) is the orthogonal projection from \({\mathcal L}^{2,1}\) onto \({\mathcal D}_h\). It is known that every function in \({\mathcal L}^{2,1}\) admits its boundary function defined a.e. via radial limits. First, the authors characterize boundedness/compactness of Toeplitz operators. Boundedness is characterized by means of the boundedness of the Poisson extension of the boundary function of the symbol and certain Carleson conditions associated with the holomorphic Dirichlet space, and compactness is characterized simply by the boundary vanishing property of the symbol. Next, the authors characterize when two Toeplitz operators commute. Their characterization is divided into three subcases, depending on whether or not each symbol satisfies a certain condition. Finally, the authors characterize when the product of two Toeplitz operator is another Toeplitz operator. As an application, they show that an operator of the form \(T_UT_V-T_H\) is compact if and only if it is of finite rank.