Linear Analysis Seminar

Date: August 1, 2016

Time: 3:00PM - 4:00PM

Location: BLOC 220

Speaker: Igor Klep, University of Auckland

Title: Domains of noncommutative rational functions and their matrix pencil realizations

Abstract: A noncommutative (nc) rational function r is defined through evaluations of formal nc rational expressions on tuples of matrices, giving rise to its domain r. Nc rational functions arise and are used e.g. in automata theory, algebraic combinatorics, control theory, free analysis and free probability. They form a skew field, the so-called universal skew field of fractions of a free algebra, and were first constructed by Amitsur and later studied by Cohn and many others. In this talk we shall discuss domains of nc rational functions. We will characterize regular functions whose domain is everything, and explain when two domains are comparable. This in turn will allow us to describe nc rational functions with a given domain. A key tool in this pursuit are matrix pencil realizations of nc rational functions. The talk is based on joint works with Victor Vinnikov, Jurij Volcic and James Pascoe.