Toeplitz Operators Acting on True-Poly-Bergman Type Spaces of the Two-Dimensional Siegel Domain: Nilpotent Symbols

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 $c(\text{Im}{\zeta }_1,\text{Im}{\zeta }_2-\mathrm{\mid }{\zeta }_1{\mathrm{\mid }}^2)$ are called nilpotent symbols. In this work we consider symbols of the form $a(\text{Im}{\zeta }_1)b(\text{Im}{\zeta }_2-\mathrm{\mid }{\zeta }_1{\mathrm{\mid }}^2)$, where both limits $\underset{s\to 0^{+}}{\lim }b(s)$ and $\underset{s\to +\mathrm{\infty }}{\lim }b(s)$ exist, and $a(s)$ belongs to the set of piece-wise continuous functions on $\bar{\mathbb{R}}=[-\mathrm{\infty },+\mathrm{\infty }]$ and having one-side limit values at each point of a finite set $D\subset \mathbb{R}$. We prove that the $C^{\ast }$-algebra generated by all Toeplitz operators $T_{ab}$ is isomorphic to $C(\bar{\mathrm{\Pi }})$, where $\bar{\mathrm{\Pi }}=\bar{\mathbb{R}}\times {\bar{\mathbb{R}}}_{+}$ and ${\bar{\mathbb{R}}}_{+}=[0,+\mathrm{\infty }]$.