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Journal of Operator Theory

Volume 83, Issue 2, Spring 2020  pp. 447-473.

Inverse and implicit function theorems for noncommutative functions on operator domains

Authors:  Mark E. Mancuso
Author institution:Department of Mathematics and Statistics, Washington University in St. Louis, St. Louis, MO 63130, U.S.A.

Summary: Classically, a noncommutative function is defined on a graded domain of tuples of square matrices. In this note, we introduce a notion of a noncommutative function defined on a domain $\Omega \subset B(\mathcal{H})^d$, where $\mathcal{H}$ is an infinite dimensional Hilbert space. Inverse and implicit function theorems in this setting are established. When these operatorial noncommutative functions are suitably continuous in the strong operator topology, a noncommutative dilation-theoretic construction is used to show that the assumptions on their derivatives may be relaxed from boundedness below to injectivity.

Keywords: noncommutive functions, operator noncommutative functions, free analysis, inverse and implicit function theorems, strong operator topology, dilation theory

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