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Function theory on Cartan domains and the Berezin-Toeplitz symbol calculus. (English) Zbl 0657.32001
Let $$\Omega$$ be a bounded symmetric (Cartan) domain in $${\mathbb{C}}^ n$$ with $$dV$$ normalized Lebesgue measure on $$\Omega$$. Let $$H^ 2=H^ 2(\Omega,dV)$$ denote the Bergman subspace of $$L^ 2(\Omega,dV)$$ consisting of holomorphic functions and $$T_ f$$ denote the Toeplitz operator on $$H^ 2$$. The algebra Q is defined as the maximal conjugate-closed subalgebra of $$L^{\infty}(\Omega)$$ for which $$T_ fT_ g-T_{fg}$$ is a compact operator for all $$f,g$$ in $$Q$$. To characterize $$Q$$, the Berezin transform $$\tilde f$$ is introduced and the algebras $${\mathcal I}$$ and $$\tilde Q$$ are defined as the sets of $$f\in L^{\infty}$$ such that ($$| f|) \tilde{\;}(z)\to 0$$ and $$(| f|^ 2)$$ $$\tilde{\;}(z)-| \tilde f(z)|^ 2\to 0,$$ respectively,as $$z\to \partial \Omega$$. The algebra $$VO_{\partial}$$ denotes the set of bounded and continuous functions on $$\Omega$$ with vanishing oscillation at the boundary. The algebra $$VMO_{\partial}(r)$$ denotes the subalgebra of $$L^{\infty}$$ with the mean oscillation on the closed Bergman metric ball centered at z with radius r approaches 0 as $$z\to \partial \Omega$$. In this paper, the authors show the equivalence of these function spaces by proving the sequence of inclusion $$Q\subseteq \tilde Q\subseteq VMO_{\partial}(r)\subseteq VO_{\partial}+{\mathcal I}\subseteq Q.$$ At the end, the authors conjecture that the above result holds for any strictly pseudoconvex domain.
Reviewer: S.H.Tung

##### MSC:
 32A10 Holomorphic functions of several complex variables 47B38 Linear operators on function spaces (general) 32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
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