Interpolation problems for Schur-class operator-valued functions on the unit disk developed spectacularly in the 1980s-1990s due to close connections with the then emerging theory of $H^\infty$ control. In particular the Ukrainian school of operator theory (led by Potapov and followed up by work of Katsnelson-Kheifets-Yuditskii among others) formulated a formally more general problem called the Abstract Interpolation Problem and obtained an explicit procedure for obtaining a Redheffer-type linear-fractonal-transformation parametrization of the set of all solutions. Here we formulate an analogous problem for de Branges-Rovnyak space-valued functions (with de Branges-Rovnyak-space norm constraint) and show how a parametrization of the set of all solutions can be obtained via identifying a connection with a KKY Abstract Interpolation Problem. The results also have connections with earlier work concerning kernels of Toeplitz operators and on nearly invariant subspaces of the backward shift operator.

This is joint work with Vladimir Bolotnikov (William and Mary, Williamburg, USA) and Sanne ter Horst (Northwest University, Potschefstroom, South Africa).