Spectral aspects of the Berezin transform

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 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 Kahler manifolds with constant scalar curvature. Furthermore, viewing POVMs as data clouds, we study their spectral features via geometry of measure metric spaces and the diffusion distance.