@inbook{cb41f03ce61747369b05730b0e402266,

title = "Interpolation and transfer-function realization for the noncommutative schur–agler class",

abstract = "The Schur–Agler class consists of functions over a domain satisfying an appropriate von Neumann inequality. Originally defined over the polydisk, the idea has been extended to general domains in multivariable complex Euclidean space with matrix polynomial defining function as well as to certain multivariable noncommutative-operator domains with a noncommutative linear-pencil defining function. Still more recently there has emerged a free noncommutative function theory (functions of noncommuting matrix variables respecting direct sums and similarity transformations). The purpose of the present paper is to extend the Schur–Agler-class theory to the free noncommutative function setting. This includes the positive-kerneldecomposition characterization of the class, transfer-function realization and Pick interpolation theory. A special class of defining functions is identified for which the associated Schur–Agler class coincides with the contractivemultiplier class on an associated noncommutative reproducing kernel Hilbert space; in this case, solution of the Pick interpolation problem is in terms of the complete positivity of an associated Pick matrix which is explicitly determined from the interpolation data.",

keywords = "Completely positive noncommutative kernel, Noncommutative contractive-multiplier class, Noncommutative function, Noncommutative pick interpolation, Noncommutative schur–agler class",

author = "Ball, {Joseph A.} and Gregory Marx and Victor Vinnikov",

note = "Funding Information: The research of the first and third authors was partially supported by the US-Israel Binational Science Foundation. It is also a pleasure to acknowledge the contribution of Orr Shalit for penetrating discussions leading to the observations in Section 3.5. We would also like to thank the anonymous referee for a careful reading of the manuscript with many constructive comments for improving the exposition. Publisher Copyright: {\textcopyright} Springer International Publishing AG, part of Springer Nature 2018.",

year = "2018",

month = jan,

day = "1",

doi = "10.1007/978-3-319-62527-0_3",

language = "English",

series = "Operator Theory: Advances and Applications",

publisher = "Springer International Publishing",

pages = "23--116",

booktitle = "Operator Theory",

}