Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32D99, 47A55

Retrieve articles in all journals with MSC: 32D99, 47A55

Additional Information

Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society