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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

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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