Every complete Pick space satisfies the column-row property

TL;DR

This paper proves that all complete Pick spaces satisfy the column-row property, leading to significant implications for multiplier theory, interpolation, and extremal problems in related function spaces.

Contribution

It establishes that every complete Pick space satisfies the column-row property, a key structural feature with broad consequences in operator and function space theory.

Findings

Every complete Pick space satisfies the column-row property.

Applications to weak product spaces, including factorization and invariant subspaces.

A new proof characterizing interpolating sequences without relying on the Kadison-Singer solution.

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.