Coincidence of Schur multipliers of the Drury-Arveson space

TL;DR

This paper characterizes when two operator-valued Schur multipliers on the Drury-Arveson space are coincident, linking this to the unitarily coincident nature of their associated colligations, in a multi-variable operator theory context.

Contribution

It provides a new characterization of the coincidence of Schur multipliers in the multi-variable setting via colligation unitarity.

Findings

Coincidence of Schur multipliers is characterized by unitarily coincident colligations.

The results are specific to the multi-variable Drury-Arveson space setting.

Single-variable theory does not exhibit these phenomena.

Abstract

In a purely multi-variable setting (i.e., the issues discussed in this note are not interesting in the single variable operator theory setting), we show that the coincidence of two operator valued Schur class multipliers of a certain kind on the Drury-Arveson space is characterized by the fact that the associated colligations (or a variant, obtained canonically) are `unitarily coincident' in a sense to be made precise in the last section of this article.