After a review of the Markov–Krein transform in the case of a one variable, and the Hilbert space interpretation of it (the phase shift), we will specialize the Markov–Krein transform to 2D. This will bring us to a relaxation of the Heisenberg commutation relation, this time filled by bounded linear transforms. The spectral invariant of this class of so called hyponormal operators is called the principal function. It is a measurable function of compact support, carrying a degree of shade. We will sketch the main specific results pertaining to hyponormal operators.

For the rest of the lecture we will link the resulting inverse spectral problem to image processing, potential theory, Hele–Shaw flows, integrable systems, and the regularity of free boundaries. Current advances with precise open questions will be detailed.

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Last updated: 26 Aug 2024