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# Free Function Theory Through Matrix Invariants

Published:2015-12-02
Printed: Apr 2017
• Igor Klep,
Department of Mathematics, The University of Auckland, New Zealand
• Špela Špenko,
Institute of Mathematics, Physics, and Mechanics, Ljubljana, Slovenia
 Format: LaTeX MathJax PDF

## Abstract

This paper concerns free function theory. Free maps are free analogs of analytic functions in several complex variables, and are defined in terms of freely noncommuting variables. A function of $g$ noncommuting variables is a function on $g$-tuples of square matrices of all sizes that respects direct sums and simultaneous conjugation. Examples of such maps include noncommutative polynomials, noncommutative rational functions and convergent noncommutative power series. In sharp contrast to the existing literature in free analysis, this article investigates free maps \emph{with involution} -- free analogs of real analytic functions. To get a grip on these, techniques and tools from invariant theory are developed and applied to free analysis. Here is a sample of the results obtained. A characterization of polynomial free maps via properties of their finite-dimensional slices is presented and then used to establish power series expansions for analytic free maps about scalar and non-scalar points; the latter are series of generalized polynomials for which an invariant-theoretic characterization is given. Furthermore, an inverse and implicit function theorem for free maps with involution is obtained. Finally, with a selection of carefully chosen examples it is shown that free maps with involution do not exhibit strong rigidity properties enjoyed by their involution-free counterparts.
 Keywords: free algebra, free analysis, invariant theory, polynomial identities, trace identities, concomitants, analytic maps, inverse function theorem, generalized polynomials
 MSC Classifications: 16R30 - Trace rings and invariant theory 32A05 - Power series, series of functions 46L52 - Noncommutative function spaces 15A24 - Matrix equations and identities 47A56 - Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones) 15A24 - Matrix equations and identities 46G20 - Infinite-dimensional holomorphy [See also 32-XX, 46E50, 46T25, 58B12, 58C10]

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