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.

Mathematics Subject Classification (2010). Primary 47B35; Secondary 47L80, 47G10.