Commutant Lifting for Commuting Row Contractions

The commutant lifting theorem of Sz.Nagy and Foiaş [22, 21] is a central result in the dilation theory of a single contraction. It states that if T ∈ B(H) is a contraction with isometric dilation V acting on K ⊃ H, and TX = XT , then there is an operator Y with ‖Y ‖ = ‖X‖, V Y = Y V and PHY = XPH. This result is equivalent to Ando’s Theorem that two commuting contractions have a joint (power) dilation to commuting isometries. However the Ando dilation is not unique, and Varopoulos [23] showed that it fails for a triple of commuting contractions. In particular, the commutant lifting result does not generalize if one replaces T by a commuting pair T1 and T2 [14]. See Paulsen’s book [17] for a nice treatment of these issues. There is a multivariable context where the commutant lifting theorem does hold. This is the case of a row contraction T = [