Toeplitz operators on two poly-Bergman-type spaces of the Siegel domain $D_2\subset {\mathbb{C}}^2$ with continuous nilpotent symbols

We studied Toeplitz operators acting on certain poly-Bergman-type spaces of the Siegel domain $D_2\subset {\mathbb{C}}^2$. Using continuous nilpotent symbols, we described the $C^{\ast }$-algebras generated by such Toeplitz operators. 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 considered 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 $\underset{s\to +\mathrm{\infty }}{lim}b(s)$ exist, and $a$ belongs to the set of piece-wise continuous functions on $\overline{\mathbb{R}}=[-\mathrm{\infty },+\mathrm{\infty }]$ and with one-sided limits at $0$. We described certain $C^{\ast }$-algebras generated by such Toeplitz operators that turned out to be isomorphic to subalgebras of $M_n(\mathbb{C})\otimes C(\overline{\mathrm{\Pi }})$, where $\overline{\mathrm{\Pi }}=\overline{\mathbb{R}}\times {\overline{\mathbb{R}}}_{+}$ and ${\overline{\mathbb{R}}}_{+}=[0,+\mathrm{\infty }]$.