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

Volume 15, Issue 1, pp 14--40
Publication Date: May 29, 2021 Submission Date: January 05, 2021 Revision Date: March 06, 2021 Accteptance Date: March 18, 2021
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Authors

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

Abstract

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 $\mathbb{H}$.

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ISRP Style

Ilwoo Cho, Certain nonlinear functions acting on the vector space $\mathbb{H}^{n}$ over the Quaternions $\mathbb{H}$, Journal of Nonlinear Sciences and Applications, 15 (2022), no. 1, 14--40

AMA Style

Cho Ilwoo, Certain nonlinear functions acting on the vector space $\mathbb{H}^{n}$ over the Quaternions $\mathbb{H}$. J. Nonlinear Sci. Appl. (2022); 15(1):14--40

Chicago/Turabian Style

Cho, Ilwoo. "Certain nonlinear functions acting on the vector space $\mathbb{H}^{n}$ over the Quaternions $\mathbb{H}$." Journal of Nonlinear Sciences and Applications, 15, no. 1 (2022): 14--40

Keywords

• The quaternions
• q-spectral forms
• q-spectralizations
• vector spaces over the quaternions

•  20G20
•  46F30
•  46T20
•  47S10