Toeplitz Operators Acting on True-Poly-Bergman Type Spaces of the Two-Dimensional Siegel Domain: Nilpotent Symbols
We describe certain $C^*$-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 $c(\text{Im}\, ζ_{1}, \text{Im}\, ζ_{2} - |ζ_1|^{2})$ are called nilpotent symbols. In this work we consider symbols of the form $a(\text{Im}\, ζ_1) b(\text{Im}\, ζ_2 -|ζ_1|^{2})$, where both limits $\lim\limits_{s\rightarrow 0^+} b(s)$ and $\lim\limits_{s\rightarrow +\infty} b(s)$ exist, and $a(s)$ belongs to the set of piece-wise continuous functions on $\overline{\mathbb{R}}=[-\infty,+\infty]$ and having one-side limit values at each point of a finite set $D\subset \mathbb{R}$. We prove that the $C^*$-algebra generated by all Toeplitz operators $T_{ab}$ is isomorphic to $C(\overlineΠ)$, where $\overlineΠ=\overline{\mathbb{R}} \times \overline{\mathbb{R}}_+$ and $\overline{\mathbb{R}}_+=[0,+\infty]$.