A note on the commutant lifting theorem and a generalized four block problem (Q1175037)

Let \(T\) and \(T'\) be contractions on Hilbert spaces \(H\) and \(H'\) and let \(A\) be a strict contraction \((\| A\|<1)\) from \(H\) to \(H'\) which intertwine \(T\) and \(T'\): \(T'A=AT\). It is proved in this note that the set of all contractive intertwining liftings of \(A\) to the minimal isometric dilations of \(T\) and \(T'\) is in one-to-one correspondence with the set of all contractive intertwining liftings of some classical block Hankel operator (intertwining a unilateral shift and the adjoint of another unilateral shift). This generalizes some of the results on the four block problem in control theory to the setting of the commutant lifting theorem.