The Berezin transform and Ha-plitz operators. (English) Zbl 0793.47026

Peetre, Jaak

J. Oper. Theory 24, No. 1, 165-186 (1990).

The technique of “quantization” typically involves assigning to functions on a suitable manifold linear operators on a suitable Hilbert space. In this paper, the author provides an exposition of this technique as developed in the work of Berezin, especially as it relates to Hankel and Toeplitz operators on pseudoconvex domains in \(\mathbb{C}^ n\). In particular, the author considers the Berezin transform in the context of Hermitian line bundles and their associated Hilbert spaces of holomorphic sections, and he discusses the principle examples and applications thereof.
The paper is written in a subjective, but refreshing, authorial voice.

Reviewer: J.Butz (Bridgewater)

MSC:

47B35	Toeplitz operators, Hankel operators, Wiener-Hopf operators
81S99	General quantum mechanics and problems of quantization

Keywords:

Ha-plitz operators; quantization; Hankel and Toeplitz operators on pseudoconvex domains; Berezin transform; Hermitian line bundles; associated Hilbert spaces of holomorphic sections