# A matrix characterization of boundary representations of positive matrices in the Hardy space

## TL;DR

This paper investigates which measures can generate boundary representations of positive matrices in the Hardy space, providing a potential characterization through a matrix identity and exploring special cases.

## Contribution

It introduces a new potential characterization of measures that yield boundary representations of positive matrices in the Hardy space, advancing understanding of their structure.

## Key findings

- Characterization holds in several important special cases.
- Matrix identity proposed as a criterion for boundary representation.
- Provides insights into the relationship 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 introduce a potential characterization of those measures via a matrix identity and show that the characterization holds in several important special cases.