Proper analytic free maps

By J. William Helton, Igor Klep, Scott McCullough. 2010

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This paper concerns analytic free maps. These maps are free analogs of classical analytic functions in several complex variables, and are defined in terms of non-commuting variables amongst which there are no relations – they are free variables. Analytic free maps include vector-valued polynomials in free (non-commuting) variables and form a canonical class of mappings from one non-commutative domain D in say g variables to another non-commutative domain D˜ in g˜ variables. As a natural extension of the usual notion, an analytic free map is proper if it maps the boundary of D into the boundary of D˜. Assuming that both domains contain 0, we show that if f:D→D˜ is a proper analytic free map, and f(0)=0, then f is one-to-one. Moreover, if also g=g˜, then f is invertible and f−1 is also an analytic free map. These conclusions on the map f are the strongest possible without additional assumptions on the domains D and D˜.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fb4d9da180f087617ec8b3da67e3dece5f22ef58

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Foundations of Free Noncommutative Function Theory (2012, methodology): “We mention also the work of Hadwin [46] and Hadwin–Kaonga–Mathes [47], of Popescu [85, 86, 88, 89], of Helton–Klep–McCullough [55, 51, 52, 53], and of Muhly–Solel [69, 71].”
Interpolation and transfer-function realization for the noncommutative Schur-Agler class (2016, background): “Closely related approaches to and results on such a “free noncommutative function theory” can be found in the work of Voiculescu [81, 82] and Helton-Klep-McCullough [45, 46, 47].”
Pick Interpolation for free holomorphic functions (2013, background): “lization formulas for rational functions that are J-unitary on the boundary of the domain; Helton [10] in proving positive matrix-valued functions are sums of squares; and Helton, Klep and McCullough [11, 12] and Helton and McCullough [13] in the context of developing a descriptive theory of the dom”
The implicit function theorem and free algebraic sets (2014, unknown): “The equivalence of (i) and (iii) is due to Helton, Klep and McCullough [9].”
Global Holomorphic Functions in Several Noncommuting Variables (2015, unknown): “of bounded operators; Ball, Groenewald and Malakorn [10], in the context of extending realization formulas from functions of commuting operators to functions of non-commuting operators; Alpay and Kalyuzhnyi-Verbovetzkii [6] in the context of realization formulas for rational functions that are J-uni”
Noncommutative reproducing kernel Hilbert spaces (2016, background): “ependently in the work of Voiculescu and collaborators (see [49, 50]) in the context of the needs of a free-probability theory, as well as in the work of Helton-Klep-McCullough and collaborators (see [25, 26, 27, 29, 11]) on free noncommutative Linear-Matrix-Inequality relaxations and connections wi”
The inverse function theorem and the Jacobian conjecture for free analysis (2013, background): “ for the matrices in the general inverse function theorem; it merely needs to have a multiplicative identity. 2.1. Derivatives in free analysis. Helton, Klep and McCullough diﬀerentiated free maps in [7]. They obtained the formula (2.1) f X H 0 X = f(X) Df(X)[H] 0 f(X) . These types of formulas ”