Toeplitz Operators Acting on True-Poly-Bergman Type Spaces of the Two-Dimensional Siegel Domain: 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 $c(\text{Im}{\zeta }_1,\text{Im}{\zeta }_2-|{\zeta }_1{|}^2)$ are called nilpotent symbols. In this work we consider symbols of the form $a(\text{Im}{\zeta }_1)b(\text{Im}{\zeta }_2-|{\zeta }_1{|}^2)$, where both limits $\underset{s\to 0^{+}}{\lim }b(s)$ and $\underset{s\to +\infty }{\lim }b(s)$ exist, and $a(s)$ 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 $C^{\ast }$-algebra generated by all Toeplitz operators $T_{ab}$ is isomorphic to $C(\overline{\Pi })$, where $\overline{\Pi }=\overline{\mathbb{R}}\times {\overline{\mathbb{R}}}_{+}$ and ${\overline{\mathbb{R}}}_{+}=[0,+\infty ]$.

arXiv admin note: substantial text overlap with arXiv:2306.02387