The Quaternionic Moment Problem

In this paper we develop an approach to the full quaternionic moment problem. We define a hierarchy $\mathbb{H}^{k}[q^{*}, q]\subset \mathbb{H}^{k+1}[q^{*}, q]$, $k\in \mathbb{N}_{0}\cup \{\infty \}$, 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 $\mathbb{H}^{k}[q^{*}, q]$. The hermitian part of $\mathbb{H}^{\infty }[q^{*}, q]$ is the real polynomial algebra $\mathbb{R}[x_{0}, x_{1}, x_{2}, x_{3}]$. This enables us to apply real algebraic geometry (Positivstellens\"{a}tze) to the quaternionic moment problem on $\mathbb{H}^{\infty }[q^{*}, q]$.