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 D 2 ⊂ ℂ 2 . Bounded measurable functions of the form c Im ζ 1 , Im ζ 2 − ζ 1 2 are called nilpotent symbols. In this work, we consider symbols of the form a Im ζ 1 b Im ζ 2 − ζ 1 2 , where both limits lim s → 0 + b s and lim s → + ∞ b s exist, and a s 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 T a b is isomorphic to C Π ¯ , where Π ¯ = ℝ ¯ × ℝ ¯ + and ℝ ¯ + = 0 , + ∞ .