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Non-commutative holomorphic functions on operator domains.

Jim Agler1, John E McCarthy2

  • 1Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA.

European Journal of Mathematics
|December 8, 2015
PubMed
Summary
This summary is machine-generated.

This study analyzes functions of d-tuples of bounded operators in Hilbert spaces. These functions are uniformly approximable by free polynomials on balanced open sets.

Keywords:
Free holomorphic functionsIntertwining preservingNc functions

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Area of Science:

  • Functional Analysis
  • Operator Theory
  • Complex Analysis

Background:

  • Hilbert spaces are fundamental in quantum mechanics and signal processing.
  • Bounded operators are essential for describing physical systems and mathematical transformations.
  • Polynomial approximation is a key technique in numerical analysis and function theory.

Purpose of the Study:

  • To characterize functions of d-tuples of bounded operators.
  • To investigate uniform approximation by free polynomials.
  • To explore properties on balanced open sets within Hilbert spaces.

Main Methods:

  • Utilizing concepts from functional analysis.
  • Applying techniques for uniform approximation.
  • Analyzing properties of d-tuples of operators.

Main Results:

  • Characterization of specific function classes.
  • Demonstration of uniform approximability by free polynomials.
  • Identification of relevant geometric conditions (balanced open sets).

Conclusions:

  • The study provides a deeper understanding of operator functions in Hilbert spaces.
  • It establishes a connection between operator theory and polynomial approximation.
  • The findings have implications for theoretical mathematics and potentially applied fields.