Abstract
We introduce and study, in the framework of a theory of quantum Cartan domains, a q-analog of the Berezin transform on the unit ball. We construct q-analogs of weighted Bergman spaces, Toeplitz operators, and covariant symbol calculus. In studying the analytical properties of the Berezin transform we introduce also the q-analog of the SU(n,1)-invariant Laplace operator (the Laplace–Beltrami operator) and present related results on harmonic analysis on the quantum ball. These are applied to obtain an analog of one result by A. Unterberger and H. Upmeier. An explicit asymptotic formula expressing the q-Berezin transform via the q-Laplace–Beltrami operator is also derived. At the end of the article, we give an application of our results to basic hypergeometric q-orthogonal polynomials.
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