Operator theory on noncommutative domains

In this volume we study noncommutative domains $\mathcal{D}_f\subset B(\mathcal{H})^n$ generated by positive regular free holomorphic functions $f$ on $B(\mathcal{H})^n$, where $B(\mathcal{H})$ is the algebra of all bounded linear operators on a Hilbert space $\mathcal{H}$.

Each such a domain has a universal model $(W_1,\ldots, W_n)$ of weighted shifts acting on the full Fock space with $n$ generators. The study of $\mathcal{D}_f$ is close related to the study of the weighted shifts $W_1,\ldots,W_n$, their joint invariant subspaces, and the representations of the algebras they generate: the domain algebra $\mathcal{A}_n(\mathcal{D}_f)$, the Hardy algebra $F_n^\infty(\mathcal{D}_f)$, and the $C^*$-algebra $C^*(W_1,\ldots, W_n)$. A good part of this paper deals with these issues. We also introduce the symmetric weighted Fock space $F_s^2(\mathcal{D}_f)$ 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 $\mathcal{D}_f$ are introduced and used to provide an $F_n^\infty(\mathcal{D}_f)$-functional calculus for completely non-coisometric elements of $\mathcal{D}_f(\mathcal{H})$, and a free analytic functional calculus for $n$-tuples of operators $(T_1,\ldots, T_n)$ with the joint spectral radius $r_p(T_1,\ldots,T_n)<1$. Several classical results from complex analysis have analogues in our noncommutative setting of free holomorphic functions on $\mathcal{D}_f$.

We associate with each $w^*$-closed two-sided ideal $J$ of the algebra $F_n^\infty(\mathcal{D}_f)$ a noncommutative variety $\mathcal{V}_{f,J}\subset \mathcal{D}_f$. We develop a dilation theory and model theory for $n$-tuples of operators $T:=(T_1,\ldots, T_n)$ in the noncommutative domain $\mathcal{D}_f$ (resp. noncommutative variety $\mathcal{V}_{f,J}$). We associate with each such an $n$-tuple of operators a characteristic function $\Theta_{f,T}$ (resp. $\Theta_{f,T,J}$), use it to provide a functional model, and prove that it is a complete unitary invariant for completely non-coisometric elements of $\mathcal{D}_f$ (resp. $\mathcal{V}_{f,J}$). In particular, we discuss the commutative case when $T_iT_j=T_jT_i$, $i=1,\ldots,n$.

We introduce two numerical invariants, the curvature and $*$-curvature, defined on the noncommutative domain $\mathcal{D}_p$, where $p$ is positive regular noncommutative polynomial, and present some basic properties. We show that both curvatures can be express in terms of the characteristic function $\Theta_{p,T}$.

We present a commutant lifting theorem for pure $n$-tuples of operators in noncommutative domains $\mathcal{D}_f$ (resp. varieties $\mathcal{V}_{f,J}$) and obtain Nevanlinna-Pick and Schur-Carathéodory type interpolation results. We also obtain a corona theorem for Hardy algebras associated with $\mathcal{D}_f$ (resp. $\mathcal{V}_{f,J}$).

In the particular case when $f=X_1+\cdots+X_n$, we recover several results concerning the multivariable noncommutative (resp. commutative) operator theory on the unit ball $[B(\mathcal{H})^n]_1$.