Certain nonlinear functions acting on the vector space $\mathbb{H}^{n}$ over the Quaternions $\mathbb{H}$

Cho, Ilwoo ¹

¹ Dept. of Math. and Stat., 421 Ambrose Hall, St. Ambrose Univ., 518 W. Locust St., Davenport, Iowa, 52803, USA

Journal of nonlinear sciences and its applications, Tome 15 (2022) no. 1, p. 14-40.

Résumé

In this paper, we consider a certain type of nonlinear functions acting on a finite-dimensional vector space $\mathbb{H}^{n}$ over the ring $\mathbb{H}$ of all quaternions, for $n$ $\in$ $\mathbb{N}.$ Our main results show that: (i) every quaternion $ {q\in\mathbb{H}}$ is classified by its spectrum of the realization under a canonical representation on $\mathbb{C}^{2}$; (ii) each vector of $\mathbb{H}^{n}$ is classified by $\mathbb{C}^{n}$ in an extended set-up of (i); and (iii) the (usual linear) spectral analysis on the matricial ring $ {M_{n}\left(\mathbb{C}\right)}$ of all $\left(n\times n\right)$-matrices (over $\mathbb{C}$) affects some fixed point theorems for our nonlinear functions on $\mathbb{H}^{n}$. In conclusion, we study the connections between the ``linear'' spectral theory over the complex numbers $\mathbb{C}$, and fixed point theorems for ``nonlinear'' functions over $\mathbb{H}$.

Détail

DOI : 10.22436/jnsa.015.01.02

Classification : 20G20, 46F30, 46T20, 47S10
Keywords: The quaternions, q-spectral forms, q-spectralizations, vector spaces over the quaternions

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