Every complete Pick space satisfies the column-row property

Abstract

In the theory of complete Pick spaces, the column-row property has appeared in a variety of contexts. We show that it is satisfied by every complete Pick space in the following strong form: each sequence of multipliers that induces a contractive column multiplication operator also induces a contractive row multiplication operator. In combination with known results, this yields a number of consequences. Firstly, we obtain multiple applications to the theory of weak product spaces, including factorization, multipliers and invariant subspaces. Secondly, there is a short proof of the characterization of interpolating sequences in terms of separation and Carleson measure conditions, independent of the solution of the Kadison–Singer problem. Thirdly, we find that in the theory of de Branges–Rovnyak spaces on the ball, the column-extreme multipliers of Jury and Martin are precisely the extreme points of the unit ball of the multiplier algebra.