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Iterates and the boundary behavior of the Berezin transform. (English) Zbl 0989.47027
Let \(\Omega\) be a bounded domain in \(\mathbb{C}^n\). Let \(L^\infty\) be the space of all measurable bounded functions on \(\Omega\) and \(BC= BC(\Omega)= C(\Omega)\cap L^\infty\). All measurable concepts refer to the Borel sets in \(\Omega\). An operator \(B: L^\infty\to BC\) is stochastic if it is positive and unital. This is equivalent to requiring, for each \(y\in\Omega\), the existence of a (necessarily unique) probability measure \(\mu_y\) on \(\Omega\) such that \(Bf(y)= \int_\Omega f(x) d\mu_y(x)\) for all \(f\in L^\infty\). A topic of central interest is the limiting behaviour of the iterates \(B^kf\), in particular when the stochastic operator \(B\) fixes holomorphic functions: \(Bf= f\) for \(f\in H(\Omega)\). The key result on such operators is Theorem 1.4: for each \(f\in C(\overline\Omega)\), the sequence \((B^kf)_k\) converges, pointwise and uniformly on compact subsets of \(\Omega\cup \partial_p(\Omega)\), to a function \(g\in BC(\Omega\cup \partial_p(\Omega))\) such that \(Bg= g\) and \(g|_{\partial_p\Omega}= f|_{\partial_p\Omega}\). Here \(\partial_p(\Omega)\) is the set of peak points of \(\Omega\)’s ‘disk algebra’.
Let \(\mu\) be a measure \((\geq 0)\) on \(\Omega\) and let \(A^2= A^2(\Omega, \mu)\) be the corresponding Bergman space. Assume that \(A^2\) admits a reproducing kernel \(K\) such that \(K(x,x)> 0\) for all \(x\in\Omega\). The Berezin transform for \(\mu\) is defined by \(Bf(y)= \int_\Omega f(x){|K(x,y)|^2\over K(y,y)} d\mu(x)\). This defines a stochastic operator \(L^\infty\to BC\) which fixes holomorphic functions. Under certain natural assumptions, the following hold (Theorem 2.3):
\(B\) maps \(C(\overline\Omega)\) into itself; for each \(f\in C(\overline\Omega)\) there is a \(g\in C(\overline\Omega)\) such that \(\lim_kB^kf=\) uniformly on \(\overline\Omega\). In particular, given \(\phi\in \mathbb{C}(\partial\Omega)\), there is a unique ‘\(B\)-Poisson extension’ \(g\in C(\overline\Omega)\) such that \(Bg= g\) and \(g|_{\partial_p\Omega}= \phi\).
In the (more extensive) second part of this rich paper the authors look at Cartan domains \(\Omega= G/K\) in \(\mathbb{C}^n\) and ‘Berezin transforms’ as convolution operators \(B_\mu f= f*\mu\), \(\mu\) being a \(K\)-invariant probability measure on \(\Omega\) which is absolutely continuous with respect to Lebesgue measure.

47B38 Linear operators on function spaces (general)
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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)
60J05 Discrete-time Markov processes on general state spaces
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