Free biholomorphic functions and operator model theory ✩

Let f = (f1, . . . , fn) be an n-tuple of formal power series in noncommutative indeterminates Z1, . . . , Zn such that f (0) = 0 and the Jacobian det Jf (0) = 0, and let g = (g1, . . . , gn) be its inverse with respect to composition. We assume that f and g have nonzero radius of convergence and g is a bounded free holomorphic function on the open unit ball [B(H)n]1, where B(H) is the algebra of bounded linear operators an a Hilbert space H. In this paper, several results concerning the noncommutative multivariable operator theory on the unit ball [B(H)n]−1 are extended to the noncommutative domain Bf (H) := X ∈ B(H)n: g f (X) = X and f (X) 1 for an appropriate evaluation X → f (X). We develop an operator model theory and dilation theory for Bf (H), where the associated universal model is an n-tuple (MZ1, . . . , MZn ) of left multiplication operators acting on a Hilbert space H2(f ) of formal power series. All the results of this paper have commutative versions. © 2012 Elsevier Inc. All rights reserved.