Symmetric functions of two noncommuting variables

TL;DR

This paper establishes a noncommutative analogue of symmetric functions in two variables, showing how such functions can be represented via an analytic nc-map and providing a realization formula in operator terms.

Contribution

It introduces a noncommutative map and domain that generalize symmetric functions, with a realization formula linking these functions to operators on Hilbert space.

Findings

Constructed an analytic nc-map from the biball to an infinite-dimensional nc-domain.

Proved every bounded symmetric analytic nc-function can be expressed via this map.

Established a realization formula for these functions in terms of Hilbert space operators.

Abstract

We prove a noncommutative analogue of the fact that every symmetric analytic function of $(z,w)$ in the bidisc $\D^2$ 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 $\ph$ of two noncommuting variables that is analytic on the biball, there exists a bounded analytic nc-function $\Phi$ on $\Omega$ such that $\ph=\Phi\circ S$. We also establish a realization formula for $\Phi$, and hence for $\ph$, in terms of operators on Hilbert space.