The purpose of the present monograph is to introduce the reader to a free noncommutative analogue of the celebrated Sz.-Nagy-Foias theory of contractions (on Hilbert spaces) in the closed unit ball of $B(H)^n$, where $B(H)$ is the algebra of bounded linear operators on a Hilbert space $H$. The elements of this noncommutative ball are called row contractions. This book contains results about the unit ball of $B(H)^n$ and classes of noncommutative varieties, which are derived from the isometric dilation theory of row contractions, the study of the associated universal algebras and functional calculus, the noncommutative commutant lifting theorem and interpolation, the characteristic function and operator model theory on Fock spaces, and the theory of free holomorphic (resp. pluriharmonic) functions on the unit ball of $B(H)^n$. These results have contributed significantly to the development of a free noncommutative analogue of Sz.-Nagy-Foias theory and have already had important implications in some other directions of research. The book is intended for graduate students and researchers in mathematics, engineering, and physical sciences, who are interested in exploring the interaction between multivariable operator theory, operator algebras, harmonic analysis on Fock spaces, and noncommutative function theory, as well as in their applications. The book is essentially self-contained and accessible to any reader who has had a course in functional analysis that includes an introduction to operator theory and $C^*$-algebras.
