Clark theory in the Drury-Arveson space

TL;DR

This paper generalizes Clark's theory to the multivariable setting of the Drury-Arveson space, introducing noncommutative measures and characterizing invariant subspaces via quasi-extreme multipliers.

Contribution

It extends Clark's theory to the Drury-Arveson space, replacing classical measures with states on a noncommutative operator system and introducing the concept of quasi-extreme multipliers.

Findings

Development of a noncommutative analogue of Aleksandrov-Clark measures.

Characterization of invariant subspaces using quasi-extreme multipliers.

Extension of Clark theory concepts to multivariable operator theory.

Abstract

We extend the basic elements of Clark's theory of rank-one perturbations of backward shifts, to row-contractive operators associated to de Branges-Rovnyak type spaces $\mathcal H(b)$ contractively contained in the Drury-Arveson space on the unit ball in $\mathbb C^d$. The Aleksandrov-Clark measures on the circle are replaced by a family of states on a certain noncommutative operator system, and the backward shift is replaced by a canonical solution to the Gleason problem in $\mathcal H(b)$. In addition we introduce the notion of a "quasi-extreme" multiplier of the Drury-Arveson space and use it to characterize those $\mathcal H(b)$ spaces that are invariant under multiplication by the coordinate functions.