The Quaternionic Moment Problem

Abstract

In this paper we develop an approach to the full quaternionic moment problem. We define a hierarchy ${\mathbb{H}}^k[q^{\ast },q]\subset {\mathbb{H}}^{k+1}[q^{\ast },q]$, $k\in {\mathbb{N}}_0\cup \{\infty \}$, of two-sided $\mathbb{H}$-linear spaces of quaternionic polynomials which are invariant under conjugation of quaternions and determine the hermitian parts of these spaces explicitly. Using a generalization of Choquet's theorem on adapted spaces to quaternions we provide necessary and sufficient solvability criteria for the quaternionic moment problem of each space ${\mathbb{H}}^k[q^{\ast },q]$. The hermitian part of ${\mathbb{H}}^{\infty }[q^{\ast },q]$ is the real polynomial algebra $\mathbb{R}[x_0,x_1,x_2,x_3]$. This enables us to apply real algebraic geometry (Positivstellensätze) to the quaternionic moment problem on ${\mathbb{H}}^{\infty }[q^{\ast },q]$.