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 $\Omega_n = \{ \zeta \in \mathbb{C}^n : \rho (\zeta ) > 0 \}$, $\rho (\zeta ) = \mathrm{Im}(\zeta_1 ) - |\zeta^\prime |^2$. We build a family of Kähler metrics of constant holomorphic curvature whose potentials are the $\rho^\alpha$-Bergman kernels, $\alpha > -1$, (in the sense of Z. Pasternak-Winiarski [20]). We build an anti-holomorphic embedding of $\Omega_n$ in the complex projective Hilbert space $\mathbb{C}P^n(H_\alpha^2(\Omega_n))$ 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 $L^2(\Omega, \rho^\alpha)$, for any strip $\Omega \subset \mathbb{C}$ and applied in a problem of approximation by holomorphic functions. Building on a work by T. Mazur (cf. [15]) we prove the existence of a coplete orthogonal system in $H^2_\alpha (\Omega_m)$ consisting of eigenfunctions of a certain explicity defined operator $V_a$, $a \in B_n$.