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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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