Toeplitz quantization and asymptotic expansions for real bounded symmetric domains

Abstract

An analogue of the star product, familiar from deformation quantization, is studied in the setting of real bounded symmetric domains. The analogue turns out to be a certain invariant operator, which one might call star restriction, from functions on the hermitification of the domain into functions on the domain itself. In particular, we establish the usual (i.e. semiclassical) asymptotic expansion of this star restriction, and further obtain a real- variable analogue of a theorem of Arazy and Orsted concerning the analogous expansion for the Berezin transform.