Holomorphic functions on the symmetrized bidisk - realization, interpolation and extension (Q1677457)

Let \(\mathbb G\) be the symmetrized bidisc in \(\mathbb C^2\), i.e., \(\mathbb G=\{(z_1+z_2,z_1z_2): |z_1|, |z_2|<1\}\). Starting from the extension theorem of H. Cartan, in this paper, an extension to Hilbert space operator theory is obtained, namely a necessary and sufficient condition for a holomorphic function \(f\) on a neighbourhood of a subset \(V\) of the symmetrized bidisc \(\mathbb G\) to have a \(H^\infty\)-norm preserving extension to the whole \(\mathbb G\). Besides this extension theorem, two other main results are obtained: a realization theorem and a Nevanlinna-Pick interpolation theorem. The paper is organized in five sections, the main results being presented and discussed in the first one. After a necessary study of the Hardy space \(H^2(\mathbb G)\) of the symmetrized bidisc in the second section, the following sections are devoted to the proofs of the main results: realization theorem, interpolation theorem and extension theorem for holomorphic functions on the symmetrized bidisc \(\mathbb G\) in \(\mathbb C^2\)