SPHERICALLY BALANCED HILBERT SPACES OF FORMAL POWER SERIES IN SEVERAL VARIABLES. I

Motivated by theory of spherical Cauchy dual tuples, we study the spherically balanced spaces, that is, Hilbert spaces H2(β) of formal power series in the variables z1,..., zm for which $\{{\mathrm{\beta }}_{\mathrm{n}}{\}}_{\mathrm{n}\in {\mathrm{\mathbb{Z}}}_{+}^{\mathrm{m}}}$ satisfies $\sum _{\mathrm{k}=1}^{\mathrm{m}}\frac{{\mathrm{\beta }}_{\mathrm{n}+{\mathrm{\varepsilon }}_{\mathrm{i}}+{\mathrm{\varepsilon }}_{\mathrm{k}}}^{2}}{{\mathrm{\beta }}_{\mathrm{n}+{\mathrm{\varepsilon }}_{\mathrm{i}}}^{2}}=\sum _{\mathrm{k}=1}^{\mathrm{m}}\frac{{\mathrm{\beta }}_{\mathrm{n}+{\mathrm{\varepsilon }}_{\mathrm{j}}+{\mathrm{\varepsilon }}_{\mathrm{k}}}^{2}}{{\mathrm{\beta }}_{\mathrm{n}+{\mathrm{\varepsilon }}_{\mathrm{j}}}^{2}}$ for all $\mathrm{n}\in {\mathrm{\mathbb{Z}}}_{+}^{\mathrm{m}}$ and i, j = 1,..., m. The main result in this paper states that H2(β) is spherically balanced if and only if there exist a Reinhardt measure μ supported on the unit sphere ∂B and a Hilbert space H2(γ) of formal power series in one variable such that ${\Vert \mathrm{f}\Vert }_{{\mathrm{H}}^{2}\left(\mathrm{\beta }\right)}^{2}=\underset{\partial \mathrm{\mathbb{B}}}{\int }{\Vert \mathrm{f}_{\mathrm{z}}\Vert }_{{\mathrm{H}}^{2}\left(\mathrm{\gamma }\right)}^{2}\mathrm{d}\mathrm{\mu }\left(\mathrm{z}\right) \quad (\mathrm{f}\in {\mathrm{H}}^{2}\left(\mathrm{\beta }\right))$.