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

John E Herr, Palle E. T Jorgensen and Eric S Weber
ArXiv.org
05/09/2017
DOI: 10.48550/arXiv.1705.04198
url
https://doi.org/10.48550/arXiv.1705.04198View
Preprint (Author's original) This preprint has not been evaluated by subject experts through peer review. Preprints may undergo extensive changes and/or become peer-reviewed journal articles. Open Access

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.

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