Multi-variable quaternionic spectral analysis

Abstract

In this paper, we consider finite dimensional vector spaces $\mathbb{H}^n$ over the ring $\mathbb{H}$ of all quaternions. In particular, we are interested in certain functions acting on $\mathbb{H}^n$, and corresponding functional equations. Our main results show that (i) all quaternions of $\mathbb{H}$ are classified by the spectra of their realizations under representation, (ii) all vectors of $\mathbb{H}^n$ are classified by a canonical extended setting of (i), and (iii) the usual spectral analysis on the matricial ring $M_n(\mathbb{C})$ of all $(n \times n)$-matrices over the complex numbers C has close connections with certain »non-linear« functional equations on $\mathbb{H}^n$ up to the classification of (ii).