Spectral aspects of the Berezin transform

Abstract

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 incidentally confirm a prediction of Donaldson for the spectrum of the Q-operator on Kähler 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. Contents 7 Two concepts of quantum noisea We thank S. Nonnenmacher for this explanation. b Note that after renormalization, there is a missing factor of 1/2 in front of the second term of the analogous formula in [30, (1.2)].

Berezin-Toeplitz quantization and the least unsharpness principle

“…by integration by parts. In light of Example 3.3, this fact generalizes the Karabegov-Schlichenmaier expansion [22,20] for the Berezin-Toeplitz quantizations of Theorem 2.7. Furthermore, a straightforward calculation shows that the asymptotic expansion (21) together with formulas ( 24) and (10) yield…”

“…This contradicts the fact that c + (f, f ) ≤ 0 for all f ∈ C ∞ (X, R), which holds for every Berezin-Toeplitz quantization by (20).…”