An analytic set-valued selection and its applications to the corona theorem, to polynomial hulls and joint spectra
HTML articles powered by AMS MathViewer
- by Zbigniew Slodkowski PDF
- Trans. Amer. Math. Soc. 294 (1986), 367-377 Request permission
Abstract:
It is shown that for every annulus $P = \{ z \in {{\mathbf {C}}^n}:\delta < |z| < r\}$, $\delta > 0$, there exists a set-valued correspondence $z \to K(z):P \to {2^{{{\mathbf {C}}^n}}}$, whose graph is a bounded relatively closed subset of the manifold $\{ (z,w) \in P \times {{\mathbf {C}}^n}:{z_1}{w_1} + \cdots + {z_n}{w_n} = 1\}$ which can be covered by $n$-dimensional analytic manifolds. The analytic set-valued selection $K$ obtained thereby is then applied to several problems in complex analysis and spectral theory which involve solving the equation ${a_1}{x_1} + \cdots + {a_n}{x_n} = y$. For example, an elementary proof is given of the following special case of a theorem due to Oka: every bounded pseudoconvex domain in ${{\mathbf {C}}^2}$ is a domain of holomorphy.References
- H. Alexander and John Wermer, On the approximation of singularity sets by analytic varieties. II, Michigan Math. J. 32 (1985), no. 2, 227–235. MR 783577, DOI 10.1307/mmj/1029003190
- Herbert Alexander and John Wermer, Polynomial hulls with convex fibers, Math. Ann. 271 (1985), no. 1, 99–109. MR 779607, DOI 10.1007/BF01455798
- Bernard Aupetit, Analytic multivalued functions in Banach algebras and uniform algebras, Adv. in Math. 44 (1982), no. 1, 18–60. MR 654547, DOI 10.1016/0001-8708(82)90064-0
- 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. MR 769495, DOI 10.1090/conm/032/769495
- Bernard Aupetit and John Wermer, Capacity and uniform algebras, J. Functional Analysis 28 (1978), no. 3, 386–400. MR 496966, DOI 10.1016/0022-1236(78)90095-2
- 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. MR 850666
- Raghavan Narasimhan, Introduction to the theory of analytic spaces, Lecture Notes in Mathematics, No. 25, Springer-Verlag, Berlin-New York, 1966. MR 0217337 T. J. Ransford, Analytic multivalued functions, Doctoral thesis, Univ. of Cambridge, 1984.
- 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. MR 759303, DOI 10.1112/plms/s3-49.3.537
- T. J. Ransford, Interpolation and extrapolation of analytic multivalued functions, Proc. London Math. Soc. (3) 50 (1985), no. 3, 480–504. MR 779400, DOI 10.1112/plms/s3-50.3.480
- Zbigniew Słodkowski, Analytic set-valued functions and spectra, Math. Ann. 256 (1981), no. 3, 363–386. MR 626955, DOI 10.1007/BF01679703 —, Analytic multifunctions, $q$-plurisubharmonic functions and uniform algebras, Proc. Conf. Banach Algebras and Several Complex Variables (F. Greenleaf and D. Gulick, eds.), Contemp. Math., vol. 32, Amer. Math. Soc., Providence, R. I., 1984. —, Local maximum property and $q$-plurisubharmonic functions in uniform algebras, J. Math. Anal. Appl. (to appear). —, Analytic perturbations of Taylor spectrum (to appear).
- Zbigniew Slodkowski, Polynomial hulls with convex sections and interpolating spaces, Proc. Amer. Math. Soc. 96 (1986), no. 2, 255–260. MR 818455, DOI 10.1090/S0002-9939-1986-0818455-X
- Zbigniew Slodkowski, Uniform algebras and analytic multifunctions, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 75 (1983), no. 1-2, 9–18 (1984) (English, with Italian summary). MR 780802
- Joseph L. Taylor, A joint spectrum for several commuting operators, J. Functional Analysis 6 (1970), 172–191. MR 0268706, DOI 10.1016/0022-1236(70)90055-8
- 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. MR 769516, DOI 10.1090/conm/032/769516
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 294 (1986), 367-377
- MSC: Primary 32D99; Secondary 47A55
- DOI: https://doi.org/10.1090/S0002-9947-1986-0819954-1
- MathSciNet review: 819954