Interpolation and transfer-function realization for the noncommutative Schur-Agler class

By Joseph A. Ball, Gregory Marx, Victor Vinnikov. 2016

54 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/39e38c4fe71770dbc6512f8e5cbefcd5d0c0a26e

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball (2017, methodology): “We now define the nc Drury–Arveson space, which was introduced by Ball, Marx and Vinnikov in [14]; this nc reproducing Hilbert space will play a central role in this paper.”
Noncommutative reproducing kernel Hilbert spaces (2016, methodology): “btain the free noncommutative analogue of the theory of de BrangesRovnyak spaces and the de Branges-Rovnyak theory of minimal decompositions and Brangesian complementary spaces. In our followup paper [13] we use various parts of the material in the present paper to develop further the work of Agler-”