A Characterization of Boundary Representations of Positive Matrices in the Hardy Space via the Abel Product

TL;DR

This paper characterizes which measures provide boundary representations of positive matrices in the Hardy space using a new operator product called the Abel Product, advancing harmonic analysis on the unit disc.

Contribution

It introduces the Abel Product and provides a characterization of boundary measures for positive matrices in the Hardy space.

Findings

Characterization of boundary measures via a matrix identity

Introduction of the Abel Product operator

Link between spectral measures and boundary representations

Abstract

Spectral measures give rise to a natural harmonic analysis on the unit disc via a boundary representation of a positive matrix arising from a spectrum of the measure. We consider in this paper the reverse: for a positive matrix in the Hardy space of the unit disc we consider which measures, if any, yield a boundary representation of the positive matrix. We prove a characterization of those representing measures via a matrix identity by introducing a new operator product called the Abel Product.