Aleksandrov–Clark Theory for Drury–Arveson Space

Abstract

Recent work has demonstrated that Clark’s theory of unitary perturbations of the backward shift on a deBranges–Rovnyak space on the disk has a natural extension to the several-variable setting. In the several-variable case, the appropriate generalization of the Schur class of contractive analytic functions is the closed unit ball of the Drury–Arveson multiplier algebra and the Aleksandrov–Clark measures are necessarily promoted to positive linear functionals on a symmetrized subsystem of the Free Disk operator system $$\mathcal {A} _d + \mathcal {A} _d ^*$$ , where $$\mathcal {A} _d$$ is the Free or Non-commutative Disk Algebra on d generators. We continue this program for vector-valued Drury–Arveson space by establishing the existence of a canonical ‘tight’ extension of any Aleksandrov–Clark map to the full Free Disk operator system. We apply this tight extension to generalize several earlier results and we characterize all extensions of the Aleksandrov–Clark maps.