Spectral aspects of the Berezin transform

Ioos, Louis; Kaminker, Victoria; Polterovich, Leonid; Shmoish, Dor

Annales Henri Lebesgue, Volume 3 (2020), pp. 1343-1387.

We discuss the Berezin transform, a Markov operator associated to positive operator valued measures (POVMs), in a number of contexts including the Berezin–Toeplitz quantization, Donaldson’s dynamical system on the space of Hermitian inner products on a complex vector space, representations of finite groups, and quantum noise. In particular, we calculate the spectral gap for quantization in terms of the fundamental tone of the phase space. Our results confirm a prediction of Donaldson for the spectrum of the $Q$-operator on Kähler manifolds with constant scalar curvature, and yield exponential convergence of Donaldson’s iterations to the fixed point. Furthermore, viewing POVMs as data clouds, we study their spectral features via geometry of measure metric spaces and the diffusion distance.

Nous étudions la transformée de Berezin, un opérateur de Markov associé aux mesures à valeur opérateurs positifs (POVM), dans un certain nombre de contextes incluant la quantificaton de Berezin–Toeplitz, le système dynamique de Donaldson sur l’espace des produits Hermitiens d’un espace vectoriel complexe, les représentations de groupes finis, et le bruit quantique. En particulier, nous calculons le trou spectral de la quantification en termes de l’harmonique fondamentale de l’espace des phases. Nos résultats confirment une prédiction de Donaldson à propos du spectre de l’opérateur $Q$ sur les variétés de Kähler à coubure scalaire constante, et impliquent la convergence exponentielle du système dynamique de Donaldson vers son point fixe. De plus, en regardant un POVM comme un nuage de points, nous étudions ses propriétés spectrales à travers la géométrie des espaces métriques mesurés et la distance de diffusion.

Classification: 53D50, 81P15
Keywords: Berezin–Toeplitz quantization, Berezin transform, Laplace–Beltrami operator, balanced metric, Positive Operator Valued Measure

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