On the general intertwining lifting problem. I

Abstract

We consider the general intertwining lifting problem as formulated in [F1] and which is connected to interpolation problems in reproducing kernel Hilbert spaces. We reduce this general problem to the case where the operators involved are nn block upper-triangular. As a consequence, we show that the causal commutant lifting (see [FT]) and the general intertwining lifting (or extension) problems are equivalent. We also obtain a seemingly new commutant lifting result for the case where one of the operators involved is nilpotent and the other canonical block Jordan. Finally, as an application, we obtain a completely new proof for the Ceausescu--Carswell--Schubert result (see [Ce], [CaS]).