Free biholomorphic functions and operator model theory

Let f=(f1,…,fn)f=(f1,…,fn) be an n  -tuple of formal power series in noncommutative indeterminates Z1,…,ZnZ1,…,Zn such that f(0)=0f(0)=0 and the Jacobian detJf(0)≠0, and let g=(g1,…,gn)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 [Bn(H)]1[B(H)n]1, where B(H)B(H) is the algebra of bounded linear operators an a Hilbert space HH. In this paper, several results concerning the noncommutative multivariable operator theory on the unit ball [B(H)n]1− are extended to the noncommutative domainBf(H):={X∈Bn(H):g(f(X))=X and ‖f(X)‖⩽1}Bf(H):={X∈B(H)n:g(f(X))=X and ‖f(X)‖⩽1} for an appropriate evaluation X↦f(X)X↦f(X). We develop an operator model theory and dilation theory for Bf(H)Bf(H), where the associated universal model is an n  -tuple (MZ1,…,MZn)(MZ1,…,MZn) of left multiplication operators acting on a Hilbert space H2(f)H2(f) of formal power series. All the results of this paper have commutative versions.