On the representation of holomorphic functions on polyhedra (Q2443021)

The authors show how operator theory ideas can be adapted to obtain precise bounds for Oka's theorem [\textit{K. Oka}, J. Sci. Hiroshima Univ., Ser. A 6, 245--255 (1936; Zbl 0015.30903)] which gives a representation for holomorphic functions defined on \(p\)-polyhedra in \(\mathbb{C}^d\). The proofs rely on the existence of realizations. Given a collection \(\delta_1,\dots, \delta_m\) of polynomials in \(d\) variables, define \[ K_\delta = \{\lambda\in\mathbb{C}^d:\;|\delta_j(\lambda)|\leq 1,\;j=1,\dots, m\}. \] An operator analog of \(K_\delta\) is \[ \mathcal{F}_\delta = \{T:\;\|\delta_j(T)\|\leq 1,\;j=1,\dots, m\}, \] where \(T= (T_1,\dots, T_d)\) is a \(d\)-tuple of pairwise commuting operators, acting on a complex Hilbert space. If \(\phi\) is holomorphic on a neighborhood of \(K_\delta\), then the following quantities are correctly defined: \[ \|\phi\|_\delta^+ = \sup_{T\in\mathcal{F}_\delta} \|\phi(T)\|; \quad \rho(\phi) = \lim_{t\to 1-} \|\phi\|_{t\delta}^+. \] The authors prove that if \(\Phi\) is holomorphic on a neighborhood of the closed polydisc \((\mathbb{D}^-)^m\) and if \(\phi= \Phi\circ\delta\), then \[ \rho(\phi) \leq \|\Phi\|_m, \] where \[ \|\Phi\|_m = \sup\{\|\Phi(\mathcal{C})\|:\;\mathcal{C} \text{\;is an \(m\)-tuple of pairwise commuting contractions},\;\sigma(\mathcal{C})\subseteq\mathbb{D}^m\}. \] Furthermore, if \(\varepsilon>0\) then there exits a \(\Phi\) holomorphic on a neighborhood of \((\mathbb{D}^-)^m\) such that \(\phi=\Phi\circ\delta\) and \(\|\Phi\|_m< \rho(\phi)+\varepsilon\).