A remark on two solutions to the rational Nehari interpolation problem (Q1175039)

For Hilbert spaces \(E\) and \(E'\), let \(H\) be the Hankel operator from \(H^ 2(E)\) to \(L^ 2(E')\ominus H^ 2(E')\) defined by \(Hf=P_ -(Q_ 0f)\) for \(f\) in \(H^ 2(E)\), where \(Q_ 0\) is a fixed rational function in \(L^ \infty(E,E')\) with only negative Fourier coefficients, and \(P_ -\) denotes the orthogonal projection onto \(L^ 2(E')\ominus H^ 2(E')\). Under the assumption \(\| H\|<1\), the Nehari interpolation problem asks to find all \(\psi\) in \(H^ \infty(E,E')\) such that \(\| Q_ 0+\psi \|_ \infty\leq 1\). (This problem is equivalent to finding the set of all contractive intertwining liftings of \(H\).) In the book ``The commutant lifting approach to interpolation problems'' (1990; Zbl 0718.47010), \textit{C. Foiaş} and \textit{A. E. Frazho} presented two different methods (Procedures XIII.9.1 and XIV.9.4 in the book), both based on the Schur representation for the commutant lifting theorem, to solve this problem. In the present paper, the authors show that the solutions of these methods can be expressed in terms of a minimal realization for the function \(G(z)=(1/z)Q_ 0(1/z)\) and its associated controllability and observability grammians so that they are more akin to the formulas obtained before by other people.