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 ⊂ 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 , + ∞ .

Parametric rigidity of real families of conformal diffeomorphisms tangent to x→−x

We prove that one-parameter families of real germs of conformal diffeomorphisms tangent to the involution x ↦−x are rigid in the parameter. We establish a connection between the dynamics in the Poincaré and Siegel domains. Although repeatedly employed in the literature, the dynamics in the Siegel domain does not explain the intrinsic real properties of these germs. Rather, these properties are fully elucidated in the Poincaré domain, where the fixed points are linearizable. However, a detailed study of the dynamics in the Siegel domain is of crucial importance. We relate both points of view on the intersection of the Siegel normalization domains.

KORÁNYI’S LEMMA FOR HOMOGENEOUS SIEGEL DOMAINS OF TYPE II. APPLICATIONS AND EXTENDED RESULTS

We show that the modulus of the Bergman kernel $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}B(z, \zeta )$ of a general homogeneous Siegel domain of type II is ‘almost constant’ uniformly with respect to $z$ when $\zeta $ varies inside a Bergman ball. The control is expressed in terms of the Bergman distance. This result was proved by A. Korányi for symmetric Siegel domains of type II. Subsequently, R. R. Coifman and R. Rochberg used it to establish an atomic decomposition theorem and an interpolation theorem by functions in Bergman spaces $A^p$ on these domains. The atomic decomposition theorem and the interpolation theorem are extended here to the general homogeneous case using the same tools. We further extend the range of exponents $p$ via functional analysis using recent estimates.