or an analog of a Blaschke product for a Hilbert space with Schwarz–Pick kernel (this is a wider class than the class of Hilbert spaces with Nevanlinna–Pick kernel), it is proved that only finitely many elementary multipliers may have zeros on a fixed compact set. It is also proved that partial Blaschke products multiplied by an appropriate reproducing kernel converge in the Hilbert space. These abstract theorems are applied to the weighted Hardy spaces in the unit disk and to the Drury–Arveson spaces.