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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Berezin Quantization and Reproducing Kernels
on Complex Domains

Author: Miroslav Englis
Journal: Trans. Amer. Math. Soc. 348 (1996), 411-479
MSC (1991): Primary 46N50, 32A07; Secondary 46E22, 32C17, 32H10, 81S99
MathSciNet review: 1340173
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Abstract: Let $\Omega $ be a non-compact complex manifold of dimension $n$, $\omega =\partial \overline{\partial }\Psi $ a Kähler form on $\Omega $, and $ K_\alpha ( x,\overline{y})$ the reproducing kernel for the Bergman space $ A^2_\alpha $ of all analytic functions on $\Omega $ square-integrable against the measure $e^{-\alpha \Psi } |\omega ^n|$. Under the condition

\begin{equation*}K_\alpha ( x,\overline{x})= \lambda _\alpha e^{\alpha \Psi (x)} \end{equation*}

F. A. Berezin [Math. USSR Izvestiya 8 (1974), 1109--1163] was able to establish a quantization procedure on $(\Omega ,\omega )$ which has recently attracted some interest. The only known instances when the above condition is satisfied, however, are just $\Omega = \mathbf{C} ^n$ and $\Omega $ a bounded symmetric domain (with the euclidean and the Bergman metric, respectively). In this paper, we extend the quantization procedure to the case when the above condition is satisfied only asymptotically, in an appropriate sense, as $\alpha \to +\infty $. This makes the procedure applicable to a wide class of complex Kähler manifolds, including all planar domains with the Poincaré metric (if the domain is of hyperbolic type) or the euclidean metric (in the remaining cases) and some pseudoconvex domains in $ \mathbf{C}^n$. Along the way, we also fix two gaps in Berezin's original paper, and discuss, for $\Omega $ a domain in $ \mathbf{C}^n$, a variant of the quantization which uses weighted Bergman spaces with respect to the Lebesgue measure instead of the Kähler-Liouville measure $|\omega ^n|$.

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Additional Information

Miroslav Englis
Email: englis@csearn.bitnet

Keywords: Khler manifolds, quantization, Berezin transform, weighted Bergman spaces, covariant symbols of operators, reproducing (Bergman) kernels, asymptotic behaviour, pseudoconvex domains, complex ellipsoids, Khler-Einstein metric
Received by editor(s): March 15, 1995
Article copyright: © Copyright 1996 American Mathematical Society