M ar 2 00 7 Operator theory on noncommutative domains

In this volume we study noncommutative domains D f ⊂ B(H) n generated by positive regular free holomorphic functions f on B(H) n , where B(H) is the algebra of all bounded linear operators on a Hilbert space H. Each such a domain has a universal model (W 1 ,. .. , Wn) of weighted shifts acting on the full Fock space with n generators. The study of D f is close related to the study of the weighted shifts W 1 ,. .. , Wn, their joint invariant subspaces, and the representations of the algebras they generate: the domain algebra A good part of this paper deals with these issues. We also introduce the symmetric weighted Fock space F 2 s (D f) and show that it can be identified with a reproducing kernel Hilbert space. The algebra of all its " analytic " multipliers will play an important role in the commutative case. Free holomorphic functions, Cauchy transforms, and Poisson transforms on noncommutative domains D f are introduced and used to provide an F ∞ n (D f)-functional calculus for completely non-coisometric elements of D f (H), and a free analytic functional calculus for n-tuples of operators (T 1 ,. .. , Tn) with the joint spectral radius rp(T 1 ,. .. , Tn) < 1. Several classical results from complex analysis have analogues in our noncommutative setting of free holomorphic functions on D f. We associate with each w *-closed two-sided ideal of the Hardy algebra F ∞ n (D f) a noncommutative variety V f,J ⊂ D f. We develop a dilation theory and model theory for n-tuples of operators T := (T 1 ,. .. , Tn) in the noncom-mutative domain D f (resp. noncommutative variety V f,J). We associate with each such an n-tuple of operators a characteristic function Θ f,T (resp. θ f,T,J), use it to provide a functional model, and prove that it is a complete unitary invariant for completely non-coisometric elements of D f (resp. V f,J). In particular , we discuss the commutative case when T i T j = T j T i , i = 1,. .. , n. We introduce two numerical invariants, the curvature and *-curvature, defined on the noncommutative domain Dp, where p is positive regular non-commutative polynomial, and present some basic properties. We show that both curvatures can be express in terms of the …