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Polynomial hulls with convex sections and interpolating spaces. (English) Zbl 0588.32017
Für ein Kompaktum \(L\subset \partial D\times\mathbb C^ m\), für das alle Fasern \(L(z)=\{w\in\mathbb C^ m: (z,w)\in L\},\) \(z\in \partial D\), nicht leere konvexe Kompakta sind, wird die polynomkonvexe Hülle \(K\) beschrieben; \(D\) bezeichne hier den Einheitskreis der komplexen Ebene. Enthält \(K\) Punkte über \(D\), so wird gezeigt, daß \(K\setminus (\partial D\times\mathbb C^ m)\) durch Graphen beschränkter analytischer Abbildungen \(f: D\to\mathbb C^ m\) ausgeschöpft werden kann. Ohne die Voraussetzung an die Fasern enthält \(K\setminus (\partial D\times\mathbb C^ m)\) i.A. keine analytischen Scheiben [vgl. J. Wermer, Ark. Mat. 20, 129–135 (1982; Zbl 0491.32013)].
Der hier gegebene Beweis beruht auf der Interpolationstheorie für Familien endlich-dimensionaler normierter Räume [vgl. R. Coifman, R. Rochberg, G. Weiss, M. Cwikel und Y. Sagher, Lect. Notes Math. 779, 123–153 (1980; Zbl 0427.46049)]. Ein anderer Beweis für \(m=1\) findet sich in der Arbeit von H. Alexander und J. Wermer [Math. Ann. 271, 99–109 (1985; Zbl 0538.32011)].

32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
46M35 Abstract interpolation of topological vector spaces
Full Text: DOI
[1] Herbert Alexander and John Wermer, On the approximation of singularity sets by analytic varieties, Pacific J. Math. 104 (1983), no. 2, 263 – 268. · Zbl 0543.32005
[2] Herbert Alexander and John Wermer, Polynomial hulls with convex fibers, Math. Ann. 271 (1985), no. 1, 99 – 109. · Zbl 0538.32011 · doi:10.1007/BF01455798 · doi.org
[3] Bernard Aupetit, Analytic multivalued functions in Banach algebras and uniform algebras, Adv. in Math. 44 (1982), no. 1, 18 – 60. · Zbl 0486.46041 · doi:10.1016/0001-8708(82)90064-0 · doi.org
[4] R. R. Coifman, R. Rochberg, G. Weiss, M. Cwikel, and Y. Sagher, The complex method for interpolation of operators acting on families of Banach spaces, Euclidean harmonic analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979) Lecture Notes in Math., vol. 779, Springer, Berlin, 1980, pp. 123 – 153. · Zbl 0422.46032
[5] T. J. Ransford, Analytic multivalued functions, Ph.D. Thesis, University of Cambridge, 1983. · Zbl 0535.30035
[6] Zbigniew Słodkowski, Analytic set-valued functions and spectra, Math. Ann. 256 (1981), no. 3, 363 – 386. · Zbl 0452.46028 · doi:10.1007/BF01679703 · doi.org
[7] -, Analytic multifunctions, \( q\)-plurisubharmonic functions and uniform algebras (Proc. Conf. Banach algebras and several complex variables), F. Greenleaf and D. Gulick, editors, Contemp. Math., vol. 32, Amer. Math. Soc., Providence, R. I., 1984, pp. 243-258.
[8] -, A generalization of Vesentini and Wermer’s theorems. Rend. Sem. Mat. Univ. Padova (to appear). · Zbl 0629.47012
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