Operators induced by certain hypercomplex systems

TL;DR

This paper explores operator representations of hypercomplex systems on Hilbert spaces, analyzing their algebraic, spectral, and probabilistic properties to deepen understanding of their mathematical structure.

Contribution

It introduces a framework for representing hypercomplex numbers as operators on Hilbert spaces and studies their properties across various mathematical perspectives.

Findings

Operators exhibit specific spectral properties.

Hypercomplex systems can be realized as 2x2 matrices.

The study reveals connections to free probability theory.

Abstract

In this paper, we consider natural Hilbert-space representations $\left\{ \left(\mathbb{C}^{2},\pi_{t}\right)\right\} _{t\in\mathbb{R}}$ of the hypercomplex system $\left\{ \mathbb{H}_{t}\right\} _{t\in\mathbb{R}}$, and study the realizations $\pi_{t}\left(h\right)$ of hypercomplex numbers $h\in\mathbb{H}_{t}$, as $\left(2\times2\right)$-matrices acting on $\mathbb{C}^{2}$, for an arbitrarily fixed scale $t\in\mathbb{R}$. Algebraic, operator-theoretic, spectral-analytic, and free-probabilistic properties of them are considered.