Absolute continuity for commuting row contractions

TL;DR

This paper characterizes absolutely continuous commuting row contractions using measure theory, showing that non-unitary cases are always absolutely continuous and exploring refinements related to ideals.

Contribution

It provides a complete measure-theoretic characterization of absolutely continuous commuting row contractions and extends the understanding of non-unitary cases.

Findings

Absolutely continuous commuting row contractions admit a weak-* continuous functional calculus.

Non-unitary row contractions are necessarily absolutely continuous.

Refinements are considered for contractions annihilated by specific ideals.

Abstract

Absolutely continuous commuting row contractions admit a weak-$*$ continuous functional calculus. Building on recent work describing the first and second dual spaces of the closure of the polynomial multipliers on the Drury-Arveson space, we give a complete characterization of these commuting row contractions in measure theoretic terms. We also establish that completely non-unitary row contractions are necessarily absolutely continuous, in direct parallel with the case of a single contraction. Finally, we consider refinements of this question for row contractions that are annihilated by a given ideal.