Interpolation in the bidisk
Let \(\Delta^k\) be the open unit polydisk in \(\mathbb{C}^k\) and \(\alpha= (\alpha_1, \dots, \alpha_n)\) an \(n\)-tuple of distinct points in \(\Delta^k\). The corresponding pick body \(P_\alpha\) is the set of all points \(\omega\in \mathbb{C}^n\) such that there is a bounded holomorphic function \(F\) in \(\Delta^k\) into the property \(|F |_\infty\leq 1\), \(F(\alpha_j) = \omega_j\) for \(1\leq j\leq n\). It is known that when \(k=1\), \(P_\alpha\) can be characterized by \[ P_\alpha = \left\{\omega\in \mathbb{C}^n: \text{ the matrix } \left[ {1-\omega_i \overline\omega_j \over 1-\alpha_i \overline\alpha_j} \right] \text{ is positive semi definite} \right\}. \] By \textit{D. Sarason} [Trans. Am. Math. Soc. 127, 179-203 (1967; Zbl 0145.39303)] it is shown also for \(k=1\) that \(P_\alpha\) can be characterized as the unit ball of a certain algebra of linear operators in \(\mathbb{C}^n\). The authors find that a similar (but not the same) result holds for the case of the bidisk, that is, \(k=2\).
scientific article; zbMATH DE number 933304