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Asymptotic expansions of Berezin transforms. (English) Zbl 0968.32013
Let $$\Omega$$ be an irreducible bounded symmetric (i.e. Cartan) domain in its Harish-Chandra realization, so that its Bergman kernel with respect to the normalized Lebesgue measure $$dm$$ is equal to $$h(x,y)^ {-p}$$, where $$h$$ is the Jordan determinant and $$p$$ the genus of $$\Omega$$. For $$\nu>p-1$$, the weighted Bergman spaces $$L^2_{\text{hol}} (\Omega,h(z,z)^{\nu-p}dm(z))$$ are nontrivial and their reproducing kernels are equal to $$c_\nu h(x,y)^{-\nu}$$, with some constant $$c_\nu>0$$. Let $$B_\nu$$ be the operator of convolution with the probability measure $$d\mu_\nu(z)=c_\nu h(z,z)^{\nu-p}dm(z)$$, i.e. $$B_\nu f(z) = \int_\Omega f\circ\phi ,d\mu_\nu$$ for any holomorphic automorphism $$\phi$$ of $$\Omega$$ taking 0 into $$z$$. $$B_\nu$$ is usually called the Berezin transform, and its asymptotic expansion $$B_\nu=\sum_{j=0}^\infty Q_j \nu^{-j}$$ as $$\nu\to\infty$$, with differential operators $$Q_j$$ satisfying $$Q_0=I$$ and $$Q_1=\Delta$$, the invariant Laplacian on $$\Omega$$, is crucial for certain quantization procedures on $$\Omega$$ [F. A. Berezin, Math. USSR Izv. 8, 1109-1165 (1974; Zbl 0312.53049)]. In the paper under review, the authors exhibit an asymptotic expansion of a slightly different kind, namely with the negative powers $$\nu^{-j}$$ replaced by the reciprocals $$1/(\nu)_{\mathbf m}$$ of the multi-Pochhammer symbols $$(\nu)_{\mathbf m}=(\nu)_{m_1} (\nu-\frac a2)_{m_2}\dots(\nu-\frac{r-1}2 a)_{m_r}$$, where $$a$$ is the characteristic multiplicity of $$\Omega$$, $$r$$ its rank, $$(\lambda)_j=\lambda(\lambda+1)\dots(\lambda+j-1)$$ the ordinary Pochhammer symbol, and $$\mathbf m$$ ranges over all nonnegative signatures, i.e. $$r$$-tuples of integers $$(m_1,\dots,m_r)$$ satisfying $$m_1\geq m_2\geq\dots\geq m_r\geq 0$$. More specifically, they show that $$B_\nu =\sum_{\mathbf m} \mathcal L_{\mathbf m}/(\nu)_{\mathbf m}$$, where $$\mathcal L_{\mathbf m}$$ are invariant differential operators on $$\Omega$$ uniquely determined by the property that $$\mathcal L_{\mathbf m}f(0)= [K_{\mathbf m}(\partial,\overline\partial)f](0)$$, where $$K_{\mathbf m} (x,\overline y)$$ is the reproducing kernel of the Peter-Weyl space corresponding to the signature $$\mathbf m$$, with respect to the Fock norm, and $$K_{\mathbf m}(\partial,\overline\partial)$$ is the differential operator obtained from $$K_{\mathbf m}(x,\overline y)$$ upon substituting $$\partial/\partial z_j$$ for $$x_j$$ and $$\partial/\partial\overline z_j$$ for $$\overline y_j$$. Similar results are also obtained for the Cartan domain $$\Omega$$ replaced by the complex $$n$$-space $$\mathbf C^n$$.
Finally, the authors also consider the asymptotics as $$\nu\to\infty$$ of the weighted Bergman projections $$P_\nu$$ of $$L^2(\Omega,d\mu_\nu)$$ onto $$L^2_{\text{hol}}(\Omega,d\mu_\nu)$$.

MSC:
 32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) 46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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