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An analytic set-valued selection and its applications to the Corona theorem, to polynomial hulls and joint spectra. (English) Zbl 0594.32008
Let $$G\subset {\mathbb{C}}^ k$$ be open. An upper semicontinuous set-valued function (multifunction) $$z\to K(z): G\to 2^{{\mathbb{C}}^ n}$$ is called analytic if for every $$(n+1)$$-dimensional complex plane $$L\subset {\mathbb{C}}^{k+n}$$, the intersection $$Y=L\cap \{(z,w): z\in G,\quad w\in K(z)\}$$ has local maximum property in the sense: there does not exist a function f(z,w) analytic in a neighbourhood of a point $$(z^*,w^*)$$ such that $$| f| |_ Y$$ has strict local maximum in $$(z^*,w^*)$$. It is shown that for every annulus $$P=\{z\in {\mathbb{C}}^ n: 0<\delta <| z| <r\}$$, there exists a multifunction $$z\to K(z): P\to 2^{{\mathbb{C}}^ n}$$ such that $$K(z)\subset L_ z=\{w\in {\mathbb{C}}^ n: z_ 1w_ 1+...+z_ nw_ n=1\}$$ for every $$z\in P$$, which is analytic. Then some applications are given: 1) an elementary proof of the following result (Oka): Every bounded pseudoconvex domain in $${\mathbb{C}}^ 2$$ is a domain of holomorphy; 2) a proof of the Corona theorem for n generators; 3) approximation of analytic multifunctions by projections of polynomial hulls; 4) analytic perturbations of the Taylor spectrum.
Reviewer: V.Brînzănescu

##### MSC:
 32A30 Other generalizations of function theory of one complex variable 30G30 Other generalizations of analytic functions (including abstract-valued functions) 32D05 Domains of holomorphy 32E30 Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs 32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables 47A10 Spectrum, resolvent
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