Toeplitz operators and algebras on Dirichlet spaces. (Q1848453)

Let \(D\subset \mathbb{C}\) be the unit disk, \(dA(z)=\frac{dxdy}{\pi}\) and let \(L^{2,1}\) denote the Sobolev space consisting of functions \(u\) such that \(\int_{D} \left(| \frac{\partial u}{\partial z}| ^2+| \frac{\partial u}{\partial \overline z}| ^2 +| u| ^2\right)dA <\infty\). By \(\mathcal D\) denote the Dirichlet space, that is, the subspace of analytic functions in \(L^{2,1}\). For a symbol \(\varphi \in C^1(D)\), the Toeplitz operator is defined as \(T_a=P(\varphi f), f \in \mathcal D,\) where \(P\) is an orthogonal projection from \(L^{2,1}\) onto \(\mathcal D\). The author studies the automorphism group of the Toeplitz \(C^\ast\)-algebra. He also shows that such Toeplitz operators have connected spectra and describes some algebraic properties of Toeplitz operators with \(C^1\)-symbols.