Toeplitz operators on the Dirichlet space (Q5916130)

Let \(D\) be the unit disk in \(\mathbb{C}\) equipped by the normalized area measure \(dA\). The Dirichlet space \(\mathcal{D}^2\) is the subspace of all holomorphic functions in the Hilbert space \(\mathcal{W}_2^1\) being the completion of the space of smooth functions \(f\) on \(D\) with the norm \[ \| f\| =\left\{\left| \int_Df\,dA\right| ^2+\int_D\big(| \partial_zf| ^2+| \partial_{\bar{z}}f| ^2\big)dA\right\}^{1/2}. \] Let \(\Omega=\big\{u\in C^1(D): u,\partial_zu,\partial_{\bar{z}}u\in L^\infty(D,dA)\big\}\). The Toeplitz operator \(T_u\) with symbol \(u\in\Omega\) is defined by \(T_uf=Q(uf)\) for \(f\in\mathcal{D}^2\), where \(Q\) is the orthogonal projection of \(\mathcal{W}_2^1\) onto \(\mathcal{D}^2\) given by \[ (Q\psi)(z)=\int_D\psi\,dA+\int_D\frac{z}{1-z\overline{w}}\,\frac{\partial\psi}{\partial w}(w)\,dA(w)\quad(z\in D) \] for \(\psi\in\mathcal{W}_2^1\). If \(u\in\Omega\), then \(T_u\) is bounded on \(\mathcal{D}^2\). The following assertions are proved. For two harmonic symbols \(u,v\in\Omega\), the operators \(T_u\) and \(T_v\) commute on \(\mathcal{D}^2\) if and only if either \(u,v\) are holomorphic, or \(u,v\) and \(1\) are linearly dependent. For a harmonic symbol \(u\in\Omega\), \(T_u\) is self-adjoint on \(\mathcal{D}^2\) if and only if \(u\) is a real constant function, and \(T_u\) is an isometry on \(\mathcal{D}^2\) if and only if \(u\) is a constant function of modulus \(1\).