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.

Free outer functions in complete Pick spaces

“…A combination of Theorem 1.3 of [36] and of Theorem 1.2 of [31] implies that if a Hilbert function space has a normalized complete Pick kernel, then for every h ∈ H ⊙ H, there is a pair of functions f, g ∈ H such that h = f g and h H⊙H = f g . It turns out that subinner/free outer pairs are relevant in this context as well.…”

“…By scaling h, we may assume that h H⊙H = 1. As mentioned in the Introduction, by the factorization result of Jury and Martin and the fact that H has the column-row property with constant 1 (Theorem 1.3 of [36] and Theorem 1.2 of [31]), there exist g 1 , g 2 ∈ H with g 1 = g 2 = 1 and…”

von Neumann's inequality for row contractive matrix tuples

“…Our next goal is to prove Theorem 1.1 in general. To this end, we use a variant of the Schur algorithm, somewhat similar to the proof of the main result in [13].…”

A note on simply interpolating sequences for the Dirichlet space

“…This is a highly non trivial theorem, proved first in [2] as a consequence of the positive solution of the Kadison-Singer problem [9] and then in [8] using different methods. The first condition is usually called weak separation and the second simply bounded Grammian condition.…”