An Infinite Product of Extremal Multipliers of a Hilbert Space with Schwarz–Pick Kernel

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Given a Hilbert functional space H on a set X, H with reproducing kernel kx(y), define the distance between a point a, a ∈ X, and a subset Z, Z ⊂ X, as follows: $$ d\left(a,Z\right)=\operatorname{inf}\left\{\left.\left\Vert \frac{k_a}{\left\Vert {k}_a\right\Vert }-h\right\Vert \kern0.5em \right|h\in \overline{\mathrm{span}}\left\{\left.{k}_z\right|z\in Z\right\}\right\}. $$ A function ψa,Z is called an extremal multiplier of H if ∥ψa,Z∥ ≤ 1, ψa,Z(a) = d(a,Z), ψa,Z(z) = 0, z ∈ Z. A space H has a Schwarz–Pick kernel if for every pair (a,Z), there exists an extremal multiplier. This definition generalizes the well-known concept of a Nevanlinna–Pick kernel. For a space H with Schwarz–Pick kernel, an inequality for the function d(a,Z) is proved. This inequality generalizes the strong triangle inequality for the metric d(a, b). For a sequence of subsets $$ {\left\{{Z}_n\right\}}_{n=1}^{\infty },{Z}_n\subset X $$ , such that $$ \sum \limits_{n=1}^{\infty}\left(1-{d}^2\left(a,{Z}_n\right)\right)<\infty $$ ,it is shown that the infinite product of extremal multipliers $$ {\psi}_{a,{Z}_n} $$ converges uniformly and absolutely on any ball with radius strictly less than one in the metric d; also, the product converges in the strong operator topology of the multiplier space.