J ul 2 00 5 CONSTRAINED MULTIVARIABLE OPERATOR THEORY

We develop a dilation theory for row contractions where P is a set of noncommutative polynomials. The model n-tuple is the universal row contraction [B1,. .. , Bn] satisfying the same constraints as T , which turns out to be, in a certain sense, the maximal constrained piece of the n-tuple [S1,. .. , Sn] of left creation operators on the full Fock space on n generators. The theory is based on a class of constrained Poisson kernels associated with T and representations of the C *-algebra generated by B1,. .. , Bn and the identity. Under natural conditions on the constraints we have uniqueness for the minimal dilation. A characteristic function ΘT is associated with any (constrained) row contraction T and it is proved that I − ΘT Θ * T = KT K * T , where KT is the (constrained) Poisson kernel of T. Consequently, for pure constrained row contractions, we show that the characteristic function is a complete unitary invariant and provide a model. We show that the curvature invariant and Euler characteristic asssociated with a Hilbert module generated by an arbitrary (resp. commuting) row contraction T can be expressed only in terms of the (resp. constrained) characteristic function of T. We provide a commutant lifting theorem for pure constrained row contractions and obtain a Nevanlinna-Pick interpolation result in our setting.