An operator-theoretic approach to theorems of the Pick-Nevanlinna and Carathéodory types (Q1112308)

\textit{M. Rosenblum} and \textit{J. Rovnyak} [in Integral equations, Oper. Theory 3, 408-436 (1980; Zbl 0439.47010)] established an abstract interpolation theorem which included the classical results of Pick- Nevanlinna, Carathéodory, and Löwner as special cases. Their theorem addresses the question of when, given a vector space X, a linear operator T on X, and vectors x and y in X, there exists a holomorphic self-map w of the unit disc such that \(y=w(T)x\) (where some additional structure is provided so that \(w(T)\) makes sense). The author of the paper under review takes a somewhat different tack and provides as his main result (Lemma 2.0) necessary and sufficient conditions for the existence of a holomorphic self-map w of the unit disc satisfying \(y(t)=M(w)^*x(t)\) where now \(x(t)\) and \(y(t)\) are mappings from some point set E into the Hardy space \(H^ 2\) and where \(M(w)\) is the operator of multiplication by w on \(H^ 2\). The proof employs the commutant lifting theorem of Sz.-Nagy and Foiaş as a central tool. The author then shows how the classical results of Pick-Nevanlinna and Carathéodory as well as the more general result of Rosenblum and Rovnyak follow from this one.