An analog of the hyperbolic metric generated by Hilbert space with Schwarz–Pick kernel

It is proved that a Hilbert function space on the set $X$ with Schwarz–Pick kernel (this is a wider class than the class of Hilbert spaces with Nevanlinna–Pick kernel) generates the metric on the set $X$ – an analog of the hyperbolic metric in the unit disk. For a sequence satisfying an abstract Blaschke condition, it is proved that the associated infinite Blaschke product converges uniformly on any fixed bounded set and in the strong operator topology of the multiplier space.