  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.
