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^* $-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 $ \lim\limits_{s\rightarrow 0^+} b(s) $ and $ \lim\limits_{s\rightarrow +\infty} 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 described certain $ C^* $-algebras generated by such Toeplitz operators that turned 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] $.