Inner and characteristic functions in polydiscs

TL;DR

This paper characterizes the characteristic functions of tuples of commuting pure contractions in polydiscs, providing a concrete description of inner functions and advancing the understanding of operator invariants in multivariable settings.

Contribution

It offers a new, explicit description of characteristic functions for operator tuples in polydiscs, linking them to inner functions and expanding the theory of Hilbert function spaces.

Findings

Concrete description of characteristic functions in polydiscs

Representation of inner functions via characteristic functions

Advancement in the theory of operator invariants

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.