Toeplitz Operators on the Bergman Spaces with Pseudodifferential Defining Symbols

Abstract

We study the C ∗-algebra generated by Toeplitz operators acting on the Bergman or poly-Bergman space over the unit disk ⅅ on the complex plane, whose pseudo differential defining symbols belong to the algebra \( \mathcal{R}= \mathcal{R}(C(\overline{\mathbb{D}};)S_\mathbb{D},S^{*}_{\mathbb{D}}). \) The algebra \( \mathcal{R}\) is generated by the multiplication operators aI, where \( {a} \in C(\overline{\mathbb{D}})\), and the following two operators \(({S}_{\mathbb{D}}\varphi)(z) = -\frac{1}{\pi} \int_{\mathbb{D}} \frac{\varphi(\zeta)}{(\zeta - z)^{2}}{dv}(\zeta) \ {\rm{and}} \ ({S}_{\mathbb{D}}^{*}\varphi)(z) = -\frac{1}{\pi} \int_{\mathbb{D}} \frac{\varphi(\zeta)}{(\zeta - z)^{2}}{dv}(\zeta)\). In the Bergman space case, both algebras \(\tau (C (\overline {\mathbb{D}}))\), generated by Toeplitz operators T a with defining symbols \( {a} \in C(\overline{\mathbb{D}})\), and \(\tau (R (C (\overline {\mathbb{D}}); S_{\mathbb{D}}, S_{\mathbb{D}}^{*}))\), generated by Toeplitz operators T A with defining symbols \( A \in R (C (\overline {\mathbb{D}}); S_{\mathbb{D}}, S_{\mathbb{D}}^{*})\), consist of the same operators, and the Fredholm symbol algebra for both of them is isomorphic and isometric to \(C (\partial{\mathbb{D}})\). At the same time, their generating Toeplitz operators possess quite different properties.