Toeplitz Operators on Two Poly-Bergman-Type Spaces of the Siegel Domain $D_2\subset {\mathbb{C}}^2$ with Continuous Nilpotent Symbols

Abstract

We describe certain $C^{\ast }$-algebras generated by Toeplitz operators with nilpotent symbols and acting on a poly-Bergman type space of the Siegel domain $D_2\subset {\mathbb{C}}^2$. Bounded measurable functions of the form $\tilde{c}(\zeta )=c(\text{Im}{\zeta }_1,\text{Im}{\zeta }_2-|{\zeta }_1{|}^2)$ are called nilpotent symbols. In this work we consider symbols of the form $\tilde{a}(\zeta )=a(\text{Im}{\zeta }_1)$ and $\tilde{b}(\zeta )=b(\text{Im}{\zeta }_2-|{\zeta }_1{|}^2)$, where both limits $\underset{s\to 0^{+}}{\lim }b(s)$ and $\underset{s\to +\infty }{\lim }b(s)$ exist, and $a$ belongs to the set of piece-wise continuous functions on $\overline{\mathbb{R}}=[-\infty ,+\infty ]$ and with one-sided limits at $0$. We describe certain $C^{\ast }$-algebras generated by such Toeplitz operators that turn out to be isomorphic to subalgebras of $M_n(\mathbb{C})\otimes C(\overline{\Pi })$, where $\overline{\Pi }=\overline{\mathbb{R}}\times {\overline{\mathbb{R}}}_{+}$ and ${\overline{\mathbb{R}}}_{+}=[0,+\infty ]$.