Spectral aspects of the Berezin transform (Q2220250)

The Berezin transform \(\mathcal{B}\) is the composition ``dequantization \(\circ\) quantization'' acting on functions on a classical phase space. The paper under review concerns the spectral properties of \(\mathcal{B}\) using the fact that its spectrum is a finite set of points in the interval \([0, 1]\). More precisely, the spectrum is expressed, with the multiplicities included, in the form: \(1=\gamma _0\geq \gamma _1\geq \gamma _2\geq \dots \geq \gamma _k\geq \dots\geq 0\) and \(\gamma :=1-\gamma _1\) is called \textit{the spectral gap}. The first main result, namely Theorem 3.1, expresses the spectral gap in the setting of Berezin-Toeplitz quantization as: \(\gamma =\frac{\hbar }{4\pi }\lambda _1+\mathcal{O}(\hbar ^2)\) where \(\lambda _1\) is the first eigenvalue of the Lapace operator \(\Delta \). Section 4 deals with the spectrum of the linearization of a class of dynamical systems on the space of all Hermitian products on a given complex linear space. Section 5 presents a geometric viewpoint on positive operator-valued measures (POVMs) while the last section contains a case study of POMVs associated to irreducible unitary representations of finite groups.