Toeplitz Operators Acting on True-Poly-Bergman Type Spaces of the Two-Dimensional Siegel Domain: Nilpotent Symbols

Abstract

We describe certain C∗-algebras generated by Toeplitz operators with nilpotent symbols and acting on a poly-Bergman type space of the Siegel domain D2⊂ℂ2. Bounded measurable functions of the form cIm ζ1,Im ζ2−ζ12 are called nilpotent symbols. In this work, we consider symbols of the form aIm ζ1bIm ζ2−ζ12, where both limits lims→0+bs and lims→+∞bs exist, and as belongs to the set of piecewise continuous functions on ℝ¯=−∞,+∞ and having one-side limit values at each point of a finite set S⊂ℝ. We prove that the C∗-algebra generated by all Toeplitz operators Tab is isomorphic to CΠ¯, where Π¯=ℝ¯×ℝ¯+ and ℝ¯+=0,+∞.