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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

An analytic set-valued selection and its applications to the corona theorem, to polynomial hulls and joint spectra


Author: Zbigniew Slodkowski
Journal: Trans. Amer. Math. Soc. 294 (1986), 367-377
MSC: Primary 32D99; Secondary 47A55
MathSciNet review: 819954
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Abstract: It is shown that for every annulus $ P = \{ z \in {{\mathbf{C}}^n}:\delta < \vert z\vert < 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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0819954-1
PII: S 0002-9947(1986)0819954-1
Article copyright: © Copyright 1986 American Mathematical Society