Inner and characteristic functions in polydiscs

TL;DR

This work extends the Sz.-Nagy–Foias framework to the polydisc $\mathbb{D}^n$ by constructing Beurling quotient modules and introducing joint defect operators. It provides a concrete formula for the characteristic function $\Theta_T$ of Beurling tuples $T$ in $\mathbb{S}_n^{B}(\mathcal{H})$, proving that $\Theta_T$ is a complete unitary invariant and enabling a model that realizes $T$ on a Beurling quotient $\mathcal{Q}_{\Theta_T}$. The authors further show how inner functions on $\mathbb{D}^n$ arise from these characteristic functions, and they develop several auxiliary tools (joint commutators, truncated defects) that clarify the multivariable structure. Special cases, including $n=1$ (recovering classical theory) and $n=2$ (pairs of contractions), are connected to well-known results, while the Beurling-quotient approach yields new insights into Ahern–Clark-type phenomena and the infinite-dimensionality of Beurling quotients. Overall, the paper delivers a canonical, operator-theoretic pathway to representing inner functions on $\mathbb{D}^n$ and to understanding unitary invariants for commuting tuples in several variables.

Abstract

Characteristic functions of linear operators are analytic functions that serve as complete unitary invariants. Such functions, as long as they are built in a natural and canonical manner, provide representations of inner functions on a suitable domain and make significant contributions to the development of various theories in Hilbert function spaces. In this paper, we solve this problem in polydiscs. In particular, we present a concrete description of the characteristic functions of tuples of commuting pure contractions and, consequently, provide a description of inner functions on polydiscs.

Theorems & Definitions (69)

• Definition 1.2
• Theorem 1.3
• Definition 1.4
• Definition 1.5
• Theorem 2.1
• Definition 2.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof