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 Ω = 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 = SU(2) and K 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 Kähler 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. Contents 1 Introduction and Main Results 2 Spectrum of the Berezin Transform of an Orbit POVM 2.1 The Berezin Transform as a Convolution Operator . . . . . . .