J. E. Pascoe

The inverse function theorem and the Jacobian conjecture for free analysis

We establish an invertibility criterion for free polynomials and free functions evaluated on some tuples of matrices. We show that if the derivative is nonsingular on some domain closed with respect to direct sums and similarity, the function must be invertible. Thus, as a corollary, we establish the Jacobian conjecture in this context. Furthermore, our result holds for commutative polynomials evaluated on tuples of commuting matrices.

Regular and positive noncommutative rational functions

Call a noncommutative rational function

Free functions with symmetry

r

Derivatives of rational inner functions: geometry of singularities and integrability at the boundary

regular if it has no singularities, i.e.,

Convex Entire Noncommutative Functions are Polynomials of Degree Two or Less

r(X)

A non-commutative Julia inequality

is defined for all tuples of self-adjoint matrices

The wedge-of-the-edge theorem: edge-of-the-wedge type phenomenon within the common real boundary

X

Free Pick functions: Representations, asymptotic behavior and matrix monotonicity in several noncommuting variables

. In this article regular noncommutative rational functions

An inductive Julia-Caratheodory theorem for Pick functions in two variables

r

Representation of free Herglotz functions

are characterized via the properties of their (minimal size) linear systems realizations