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Journal of Lie Theory 15 (2005), No. 1, 027--050
Copyright Heldermann Verlag 2005

Berezin-Toeplitz Quantization on the Schwartz Space of Bounded Symmetric Domains

Miroslav Englis
Mathematical Institute, Czech Academy of Sciences, Zitna 25, 11567 Praha 1, Czech Republic

Borthwick, Lesniewski and Upmeier [Nonperturbative deformation quantization of Cartan domains, J. Funct. Anal. 113 (1993) 153--176] proved that on any bounded symmetric domain (Hermitian symmetric space of non-compact type), for any compactly supported smooth functions f and g, the product of the Toeplitz operators TfTg on the standard weighted Bergman spaces can be asymptotically expanded into a series of another Toeplitz operators multiplied by decreasing powers of the Wallach parameter ν. This is the Berezin-Toeplitz quantization.
We remove the hypothesis of compact support and show that their result can be extended to functions f, g in a certain algebra which contains both the space of all smooth functions whose derivatives of all orders are bounded and the Schwartz space. Applications to deformation quantization are also given.

Keywords: Berezin-Toeplitz quantization, bounded symmetric domain, Schwartz space.

MSC: 22E30; 43A85, 47B35, 53D55

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