Clark Measures for Rational Inner Functions

TL;DR

This paper provides a detailed analysis of Clark measures and isometries linked to two-variable rational inner functions on the bidisk, revealing their structure, support, weights, and operator properties.

Contribution

It offers a complete description of Clark measures in the (n,1) degree case, including support, weights, and conditions for unitarity of embedding operators, connecting to Agler decompositions.

Findings

Characterization of supports and weights for Clark measures

Conditions for embedding operators to be unitary

Connections to Agler decompositions and positive pluriharmonic measures

Abstract

We analyze the fine structure of Clark measures and Clark isometries associated with two-variable rational inner functions on the bidisk. In the degree (n,1) case, we give a complete description of supports and weights for both generic and exceptional Clark measures, characterize when the associated embedding operators are unitary, and give a formula for those embedding operators. We also highlight connections between our results and both the structure of Agler decompositions and study of extreme points for the set of positive pluriharmonic measures on 2-torus.