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The Berezin transform and invariant differential operators. (English) Zbl 0843.32019
The authors study the Berezin calculus of operators, particularly the Berezin transform, on any bounded, symmetric domain. They are able to express the Berezin transform in terms of certain invariant differential operators.
Along the way, the authors provide a nice introduction to reproducing kernels, Berezin theory, mathematical quantization, and analysis on bounded symmetric domains.
As well as providing a new point of view on the Berezin transform, this paper will serve as a useful resource.

32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
47N50 Applications of operator theory in the physical sciences
32A30 Other generalizations of function theory of one complex variable (should also be assigned at least one classification number from Section 30-XX)
43A85 Harmonic analysis on homogeneous spaces
Full Text: DOI
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