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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

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
Full Text: DOI
[1] H. Alexander and John Wermer, On the approximation of singularity sets by analytic varieties. II, Michigan Math. J. 32 (1985), no. 2, 227 – 235. · Zbl 0578.32021
[2] Herbert Alexander and John Wermer, Polynomial hulls with convex fibers, Math. Ann. 271 (1985), no. 1, 99 – 109. · Zbl 0538.32011
[3] Bernard Aupetit, Analytic multivalued functions in Banach algebras and uniform algebras, Adv. in Math. 44 (1982), no. 1, 18 – 60. · Zbl 0486.46041
[4] Bernard Aupetit, Geometry of pseudoconvex open sets and distribution of values of analytic multivalued functions, Proceedings of the conference on Banach algebras and several complex variables (New Haven, Conn., 1983) Contemp. Math., vol. 32, Amer. Math. Soc., Providence, RI, 1984, pp. 15 – 34.
[5] Bernard Aupetit and John Wermer, Capacity and uniform algebras, J. Funct. Anal. 28 (1978), no. 3, 386 – 400. · Zbl 0378.46045
[6] B. Berndtsson and T. J. Ransford, Analytic multifunctions, the \overline\partial -equation, and a proof of the corona theorem, Pacific J. Math. 124 (1986), no. 1, 57 – 72. · Zbl 0602.32002
[7] Raghavan Narasimhan, Introduction to the theory of analytic spaces, Lecture Notes in Mathematics, No. 25, Springer-Verlag, Berlin-New York, 1966.
[8] T. J. Ransford, Analytic multivalued functions, Doctoral thesis, Univ. of Cambridge, 1984. · Zbl 0508.46036
[9] T. J. Ransford, Open mapping, inversion and implicit function theorems for analytic multivalued functions, Proc. London Math. Soc. (3) 49 (1984), no. 3, 537 – 562. · Zbl 0526.46045
[10] T. J. Ransford, Interpolation and extrapolation of analytic multivalued functions, Proc. London Math. Soc. (3) 50 (1985), no. 3, 480 – 504. · Zbl 0535.30035
[11] Zbigniew Słodkowski, Analytic set-valued functions and spectra, Math. Ann. 256 (1981), no. 3, 363 – 386. · Zbl 0452.46028
[12] -, Analytic multifunctions, \( q\)-plurisubharmonic functions and uniform algebras, Proc. Conf. Banach Algebras and Several Complex Variables , Contemp. Math., vol. 32, Amer. Math. Soc., Providence, R. I., 1984. · Zbl 0587.32031
[13] -, Local maximum property and \( q\)-plurisubharmonic functions in uniform algebras, J. Math. Anal. Appl. (to appear). · Zbl 0646.46047
[14] -, Analytic perturbations of Taylor spectrum (to appear).
[15] Zbigniew Slodkowski, Polynomial hulls with convex sections and interpolating spaces, Proc. Amer. Math. Soc. 96 (1986), no. 2, 255 – 260. · Zbl 0588.32017
[16] Zbigniew Slodkowski, Uniform algebras and analytic multifunctions, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 75 (1983), no. 1-2, 9 – 18 (1984) (English, with Italian summary). · Zbl 0596.46047
[17] Joseph L. Taylor, A joint spectrum for several commuting operators, J. Functional Analysis 6 (1970), 172 – 191. · Zbl 0233.47024
[18] J. Wermer, Green’s functions and polynomial hulls, Proceedings of the conference on Banach algebras and several complex variables (New Haven, Conn., 1983) Contemp. Math., vol. 32, Amer. Math. Soc., Providence, RI, 1984, pp. 273 – 278.
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