Noncommutative Multivariable Operator Theory

Abstract

In this paper, we study noncommutative domains $${\mathbb{D}_f^\varphi(\mathcal{H}) \subset B(\mathcal{H})^n}$$ generated by positive regular free holomorphic functions f and certain classes of n-tuples $${\varphi = (\varphi_1, \ldots, \varphi_n)}$$ of formal power series in noncommutative indeterminates Z 1, . . . , Z n . Noncommutative Poisson transforms are employed to show that each abstract domain $${\mathbb{D}_f^\varphi}$$ has a universal model consisting of multiplication operators (M Z1, . . . , M Z n ) acting on a Hilbert space of formal power series. We provide a Beurling type characterization of all joint invariant subspaces under M Z1, . . . , M Z n and show that all pure n-tuples of operators in $${\mathbb{D}_f^\varphi(\mathcal{H})}$$ are compressions of $${M_{Z_1} \otimes I, \ldots, M_{Z_n} \otimes I}$$ to their coinvariant subspaces. We show that the eigenvectors of $${M_{Z_1}^*, \ldots, M_{Z_n}^*}$$ are precisely the noncommutative Poisson kernels $${\Gamma_\lambda}$$ associated with the elements $${\lambda}$$ of the scalar domain $${\mathbb{D}_{f,<}^\varphi(\mathbb{C}) \subset \mathbb{C}^n}$$ . These are used to solve the Nevanlinna-Pick interpolation problem for the noncommutative Hardy algebra $${H^\infty(\mathbb{D}_f^\varphi)}$$ . We introduce the characteristic function of an n-tuple $${T=(T_1, \ldots , T_n) \in \mathbb{D}_f^\varphi(\mathcal{H})}$$ , present a model for pure n-tuples of operators in the noncommutative domain $${\mathbb{D}_f^\varphi(\mathcal{H})}$$ in terms of characteristic functions, and show that the characteristic function is a complete unitary invariant for pure n-tuples of operators in $${\mathbb{D}_f^\varphi(\mathcal{H})}$$ .