Abstract
This paper concerns free function theory. Freemaps 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 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 invarianttheoretic
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 involutionfree counterparts.
Certain Rational Functions Whose Power Series Have Positive Coefficients