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

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.