The Spectrum of the Berezin transform for Gelfand pairs

Abstract: We discuss the Berezin transform, a Markov operator associated to positive-operator valued measures (POVMs). We consider the class of so-called orbit POVMs, constructed on the quotient space $\Omega = G/K$ of a compact group $G$ by its subgroup $K$. We restrict attention to the case where $(G, K)$ is a Gelfand pair and derive an explicit formula for the spectrum of the Berezin transform in terms of the characters of the irreducible unitary representations of $G$. We then specialize our results to the case study $G = \text{SU}(2)$ and $K \simeq S^1$, and find the spectra of orbit POVMs on $S^2$. We confirm previous calculations by Zhang and Donaldson of the spectrum of the standard quantization of $S^2$ coming from K\"ahler geometry. Then, we make a couple of conjectures about the oscillations in the sequence of eigenvalues, and prove them in the simplest case of second-highest weight vector. Finally, for low weights, we prove that the corresponding orbit POVMs on $S^2$ violate the axioms of a Berezin-Toeplitz quantization.