×

Formule du binôme généralisée. (Formula of generalized binomial). (French) Zbl 0673.30040

Harmonic analysis, Proc. Int. Symp., Luxembourg/Luxemb. 1987, Lect. Notes Math. 1359, 170-180 (1988).
[For the entire collection see Zbl 0661.00009.]
Let \(V=Sym(n,{\mathbb{R}})\) be the vector space of symmetric \(n\times n\) matrices (resp. \(V=Herm(n,{\mathbb{C}})\) of Hermitian matrices). The author generalizes the binomial series to the function \(x\mapsto \det (e-x)^{- \alpha}\) (\(\alpha\in {\mathbb{C}})\) on the vector space V. The development is with respect to spherical polynomials \(\Phi_{\underline m}(x)=\int_{k}\Delta_{\underline m}(k\times k^*)dk\). The functions \(\Delta_{\underline m}(x)\) are built from the principal minorants of x, and dk is normalized Haar measure on the group \(k=O(n)\) (resp. U(n)). The generalized binomial series for \(\det (e-x)^{-\alpha}\) converges absolutely if every eigenvalue \(\lambda\) of x satisfies \(-1<\lambda <1\). Let \({\mathcal D}\) be one of the following two discs: \[ {\mathcal D}=\{z\in Sym(n,{\mathbb{C}})| \quad e-z\bar z\gg 0\}\quad or\quad {\mathcal D}=\{z\in M(n,{\mathbb{C}})| \quad e-zz^*\gg 0\}. \] Let \({\mathcal H}^ 2_{\lambda}({\mathcal D})\) be the space of holomorphic functions on \({\mathcal D}\) such that \[ \| f\|^ 2_{\lambda}=c_{\lambda}\int_{{\mathcal D}}| f(z)|^ 2 \det (e-zz^*)^{\lambda -2(N\quad /n)} dz<\infty \] \((c_{\lambda}\) for normalization only).
If \(\lambda\) is an integer \(\geq 2(N/n)\) the group \({\mathcal G}\) of holomorphic automorphisms of \({\mathcal D}\) (operating transitively on \({\mathcal D})\) acts unitarily and irreducably on \({\mathcal H}^ 2_{\lambda}({\mathcal D})\). The function \(K_{\lambda}(z,\omega)=\det (e-z\omega^*)^{- \lambda}\) is a reproducing kernel for \({\mathcal H}^ 2_{\lambda}({\mathcal D})\). A series for \(K_{\lambda}\) in terms of the functions \(\Phi_{\underline m}\) is given. The values \(\lambda\) for which \(K_{\lambda}\) is of positive type are given explicitly.
Reviewer: W.Kugler

MSC:

30H05 Spaces of bounded analytic functions of one complex variable
33C55 Spherical harmonics
43A90 Harmonic analysis and spherical functions

Citations:

Zbl 0661.00009