Foundations of Free Noncommutative Function Theory

By Dmitry S. Kaliuzhnyi-Verbovetskyi, Victor Vinnikov. 2012

197 citations recorded by the source corpus.

Introduction NC functions and their difference-differential calculus Higher order nc functions and their difference-differential calculus The Taylor-Taylor formula NC functions on nilpotent matrices NC polynomials vs. polynomials in matrix entries NC analyticity and convergence of TT series Convergence of nc power series Direct summands extensions of nc sets and nc functions (Some) earlier work on nc functions Similarity invariant envelopes and extension of nc functions Bibliography Index

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b97e9877bac13f9dc48209f71bdc262f230a14b2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The implicit function theorem and free algebraic sets (2014, unknown): “The following lemma is in [9] and [11].”
Interpolation and transfer-function realization for the noncommutative Schur-Agler class (2016, methodology): “at also Z 0 0 W ∈ Ω ⇒ f Z 0 0 W = h f(Z) 0 0 f(W) i , (2.2) and • frespects similarities, i.e.: whenever Z,Ze∈ Ωn, α∈ Cn×n with αinvertible such that Ze= αZα−1, then f(Ze) = αf(Z)α−1. (2.3) Following [52], we denote the set of all nc functions from Ω into V0,nc by T (Ω;V0,nc); we note that we do not”
Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball (2017, background): “Now writing out the Taylor–Taylor power series for f around Y (see [44]), we get that, in fact, f is the identity on a noncommutative ball with center Y .”
Free Probability and Random Matrices (2014, background): “One should, however, note that the analytic theory of operator-valued free convolution lacks at the moment some of the deeper statements of the scalar-valued theory; developing a reasonable analogue of complex function theory on an operatorvalued level, addressed as free analysis, is an active area ”
Global Holomorphic Functions in Several Noncommuting Variables (2015, background): “See [9] for a general reference to nc-functions.”
Pick Interpolation for free holomorphic functions (2013, background): “Recently, Kaliuzhnyi-Verbovetskyi and Vinnikov have written a monograph [19] that gives a panoramic view of the developments in the field to date, and establishes their Taylor-Taylor formula for nc-functions.”