Complex analysis and realization theory for classes of free noncommutative functions

Abstract

A non-commutative function is a mapping defined on the disjoint union of tuples of matrices of all sizes which is graded (maps matrices of a given size to matrices of the same size), respects direct sums and similarities. In the last decade a theory of non-commutative functions has seen extensive development, by a variety of authors in a variety of topics. Two topics of particular interest in our study is free analysis and realization theory in the non-commutative multivariable setting. In the field of free analysis we are specifically interested in complex analysis for free non-commutative functions. This is a topic where one may ask whether some classical complex analysis result can be shown to be true in the setting of non-commutative functions. Our focus in this field is on real non-commutative functions. These are mappings that are graded, respect direct sums and unitary equivalence. We show that, despite the apparent limitation of unitary equivalence, these functions are in fact non-commutative functions, i.e., they also respect similarities. Real non-commutative functions occur as the real and imaginary parts of non-commutative functions. From this point of view we prove that the real and imaginary parts of a Frechet differentiable non-commutative function satisfies a non-commutative Cauchy-Riemann equation. Furthermore, we show that the converse also holds, which extends the well-known result found in the commutative setting. We conclude our investigation of real non-commutative functions by showing that the real and imaginary parts of a Frechet differentiable non-commutative function satisfy a non-commutative version of the Laplace equation. Regarding the topic of realization theory, we restrict our focus to matrix-valued non-commutative rational functions. Our main result here is a characterization of the class of matrix-Cayley-inner non-commutative rational functions on the non-commutative right poly-halfplane. This extends a result of Ball-Kaliuzhnyi-Verbovetskyi from the commutative setting to the non-commutative setting. In particular, we show that membership in the class of matrix-Cayley-inner non-commutative rational functions on the non-commutative right poly-halfplane is equivalent to a matrix-valued non-commutative function having a finite-dimensional long-resolvent nc representation or nc Bessmertnyi realization.