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

Volume 15, Issue 1, pp 14--40 http://dx.doi.org/10.22436/jnsa.015.01.02
Publication Date: May 29, 2021 Submission Date: January 05, 2021 Revision Date: March 06, 2021 Accteptance Date: March 18, 2021

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


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