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Spectral measures give rise to a natural harmonic analysis on the unit discvia a boundary representation of a positive matrix arising from a spectrum ofthe measure. We consider in this paper the reverse: for a positive matrix inthe Hardy space of the unit disc we consider which measures, if any, yield aboundary representation of the positive matrix. We introduce a potentialcharacterization of those measures via a matrix identity and show that thecharacterization holds in several important special cases.

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