An infinite product of extremal multipliers of a Hilbert space with Schwarz–Pick kernel

I. V. Videnskii

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 47, Tome 480 (2019), pp. 73-85

Résumé

In a functional Hilbert space $H$ on a set $X$ with reproducing kernel $k_x(y)$, define the distance between a point $a$, $a\in X$, and a subset $Z$, $Z\subset X$, as follows: $$ d(a,Z)=\inf\left\{\Big\|\frac{k_a}{\|k_a\|}-h\Big\|\biggm | h\in \overline{\mathrm{span}}\big\{k_z | z\in Z\big\} \right\} . $$ A function $\psi_{a,Z}$ is called an extremal multiplier of $H$ if $\|\psi_{a,Z}\|\leq 1$, $\psi_{a,Z}(a)=d(a,Z)$, $\psi_{a,Z}(z)=0$, $z\in Z$. A space $H$ has the 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 $\{Z_n\}_{n=1}^\infty$, $Z_n\subset X$, such that $\sum\limits_{n=1}^\infty\left(1-d^2(a,Z_n)\right)<\infty$, it is shown that an 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 $d$, and also converges in the strong operator topology of the multiplier space.

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