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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.

MSC:
 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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