Committee spaces and the random column-row property

TL;DR

This paper investigates the true column-row property in Hilbert spaces of power series, demonstrating that random multipliers and large random interpolation problems generally satisfy this property, indicating its robustness.

Contribution

The paper introduces a model for random multipliers and proves they satisfy the true column-row property, and shows asymptotic satisfaction in large random interpolation problems.

Findings

Random multipliers satisfy the true column-row property.

Large random interpolation problems asymptotically satisfy the property.

Naive random search is unlikely to find violations in the Drury-Arveson space.

Abstract

A committee space is a Hilbert space of power series, perhaps in several or noncommuting variables, such that $\|z^\alpha\|\|z^\beta\| \geq \|z^{\alpha+\beta}\|.$ Such a space satisfies the true column-row property when ever the map transposing a column multiplier to a row multiplier is contractive. We describe a model for random multipliers and show that such random multipliers satisfy the true column-row property. We also show that the column-row property holds asymptotically for large random Nevanlinna-Pick interpolation problems where the nodes are chosen identically and independently. These results suggest that finding a violation of the true column-row property for the Drury-Arveson space via na\"ive random search is unlikely.