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Operator theory on noncommutative domains

About this Title

Gelu Popescu, Department of Mathematics, The University of Texas at San Antonio, San Antonio, Texas 78249

Publication: Memoirs of the American Mathematical Society
Publication Year: 2010; Volume 205, Number 964
ISBNs: 978-0-8218-4710-7 (print); 978-1-4704-0578-6 (online)
Published electronically: November 23, 2009
MathSciNet review: 2643314
Keywords:Multivariable operator theory, Free holomorphic function, Noncommutative domain, Noncommutative variety, Fock space, Weighted shifts, Invariant subspace, Hardy algebra, Cauchy transform, Poisson transform, von Neumann inequality, Bohr inequality, Functional calculus, Wold decomposition, Dilation, Characteristic function, Model theory, Curvature, Commutant lifting, Interpolation
MSC: Primary 47A20; Secondary 46L07, 47A13, 47A48, 47A57, 47B37

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Table of Contents


  • Introduction
  • Chapter 1. Operator algebras associated with noncommutative domains
  • Chapter 2. Free holomorphic functions on noncommutative domains
  • Chapter 3. Model theory and unitary invariants on noncommutative domains
  • Chapter 4. Commutant lifting and applications


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 $[B(\mathcal{H})^n]_1$.

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