On the Djrbashian kernel of a Siegel domain

We establish an inversion formula for the M. M. Djrbashian & A. H. Karapetyan integral transform (cf. [6]) on the Siegel domain $Ω_n = {ζ ∈ ℂ^n : ϱ (ζ) >0} $, $ϱ(ζ) = Im(ζ_1) - |ζ'|^2$. We build a family of Kähler metrics of constant holomorphic curvature whose potentials are the $ϱ^α$-Bergman kernels, α > -1, (in the sense of Z. Pasternak-Winiarski [20] of $Ω_n$. We build an anti-holomorphic embedding of $Ω_n$ in the complex projective Hilbert space $ℂℙ(H^2_α(Ω_n))$ and study (in connection with work by A. Odzijewicz [18] the corresponding transition probability amplitudes. The Genchev transform (cf. [9]) is shown to be well defined on $L^2(Ω, ϱ^α)$, for any strip Ω ⊂ ℂ, and applied in a problem of approximation by holomorphic functions. Building on work by T. Mazur (cf. [15]) we prove the existence of a complete orthonormal system in $H^2_α(Ω_n)$ consisting of eigenfunctions of a certain explicitly defined operator $V_a$, $a ∈ B_n$.