On the Djrbashian kernel of a Siegel domain

• Autorzy: Elisabetta Barletta , Sorin Dragomir
• Języki publikacji: EN

Abstrakty

EN

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 $V_a$, $a ∈ B_n$.

Słowa kluczowe

EN

γ-Bergman kernel; reproducing kernel Hilbert space; Djrbashian kernel; transition probability amplitude; Genchev transform

Kategorie tematyczne

• 53C55: Hermitian and K\"ahlerian manifolds
• 46E22: Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
• 32A25: Integral representations; canonical kernels (Szeg\H o, Bergman, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1998
• Tom: 127
• Numer: 1
• Strony: 47-63

Daty

wydano

1998

otrzymano

1996-07-29

poprawiono

1997-05-19

Twórcy

autor

Elisabetta Barletta

• Dipartimento di Matematica, Università della Basilicata, Via N. Sauro 85, 85100 Potenza, Italy,

autor

Sorin Dragomir

• Dipartimento di Matematica, Università della Basilicata, Via N. Sauro 85, 85100 Potenza, Italy,

Bibliografia

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