# ePrints

## Symmetric functions of two noncommuting variables

Lookup NU author(s): Professor Nicholas Young

### Abstract

We prove a noncommutative analogue of the fact that every symmetric analytic function of $(z,w)$ in the bidisc $\mathbb{D}^2$ can be expressed as an analytic function of the variables $z+w$ and $zw$. We construct an analytic nc-map $S$ from the biball to an infinite-dimensional nc-domain $\Omega$ with the property that, for every bounded symmetric function $\varphi$ of two noncommuting variables that is analytic on the biball, there exists a bounded analytic nc-function $\Phi$ on $\Omega$ such that $\varphi=\Phi\circ S$. We also establish a realization formula for $\Phi$, and hence for $\varphi$, in terms of operators on Hilbert space.

Author(s): Agler J, Young NJ

Publication type: Article

Publication status: Published

Journal: Journal of Functional Analysis

Year: 2014

Volume: 266

Issue: 9

Pages: 5709-5732

Print publication date: 01/05/2014

Online publication date: 11/03/2014

Acceptance date: 19/02/2014

Date deposited: 01/05/2014

ISSN (print): 0022-1236

ISSN (electronic): 1096-0783

Notes: Freely available at http://uk.arxiv.org/abs/1307.1588

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### Funding

Funder referenceFunder name
41219London Mathematical Society grant
DMS 1068830National Science Foundation grant on Extending Hilbert Space Operators
EP/K50340X/1UK Engineering and Physical Sciences Research Council
EP/K50340X/1UK Engineering and Physical Sciences Research Council
EP/K50340X/1UK Engineering and Physical Sciences Research Council
EP/K50340X/1UK Engineering and Physical Sciences Research Council
EP/K50340X/1UK Engineering and Physical Sciences Research Council
EP/K50340X/1UK Engineering and Physical Sciences Research Council
EP/K50340X/1UK Engineering and Physical Sciences Research Council
EP/K50340X/1UK Engineering and Physical Sciences Research Council
EP/K50340X/1UK Engineering and Physical Sciences Research Council
EP/K50340X/1UK Engineering and Physical Sciences Research Council
EP/K50340X/1UK Engineering and Physical Sciences Research Council