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 $D_2\subset {ℂ}^2$ . Bounded measurable functions of the form $c\left(\mathrm{Im}{\zeta }_1,\mathrm{Im}{\zeta }_2-{\left|{\zeta }_1\right|}^2\right)$ are called nilpotent symbols. In this work, we consider symbols of the form $a\left(\mathrm{Im}{\zeta }_1\right)b\left(\mathrm{Im}{\zeta }_2-{\left|{\zeta }_1\right|}^2\right)$ , where both limits $\underset{s\to 0^{+}}{\lim }b\left(s\right)$ and $\underset{s\to +\infty }{\lim }b\left(s\right)$ exist, and $a\left(s\right)$ belongs to the set of piecewise continuous functions on $\overline{ℝ}=\left[-\infty ,+\infty \right]$ and having one-side limit values at each point of a finite set $S\subset ℝ$ . We prove that the $C^{\ast }$ -algebra generated by all Toeplitz operators $T_{ab}$ is isomorphic to $C\left(\overline{\Pi }\right)$ , where $\overline{\Pi }=\overline{ℝ}\times {\overline{ℝ}}_{+}$ and ${\overline{ℝ}}_{+}=\left[0,+\infty \right]$ .