Noncommutative Multivariable Operator Theory (Mathematical Surveys and Monographs)

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 implicatio