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It is shown that for every annulus      P = { z ∈  C n : δ  &gt; | z | &gt; r } P = \{ z \in {{\mathbf {C}}^n}:\delta &gt; |z| &gt; r\}      ,      δ  &gt; 0 \delta &gt; 0      , there exists a set-valued correspondence      z →  K ( z ) : P →  2 C n z \to K(z):P \to {2^{{{\mathbf {C}}^n}}}      , whose graph is a bounded relatively closed subset of the manifold      { ( z , w ) ∈  P ×  C n : z 1 w 1 + ⋯  + z n w n = 1 } \{ (z,w) \in P \times {{\mathbf {C}}^n}:{z_1}{w_1} + \cdots + {z_n}{w_n} = 1\}       which can be covered by      n n      -dimensional analytic manifolds. The analytic set-valued selection      K K       obtained thereby is then applied to several problems in complex analysis and spectral theory which involve solving the equation      a 1 x 1 + ⋯  + a n x n = y {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      C 2 {{\mathbf {C}}^2}       is a domain of holomorphy.

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