Some results on nonstandard quaternionic numerical ranges of matrices

Document Type : Original Paper

Authors

1 Shahid Bahonar Uniersity of Kerman, Kerman, Iran

2 Kohbanan Education, Kohbanan, Iran

Abstract

Let $\phi$ be a nonstandard involution on the set of all quaternion numbers and $\alpha$ be a quaternion such that $\phi(\alpha) = \alpha$. For any square quaternion matrix $A$, the numerical range of $A$ with respect to $\phi$ (shortly, the nonstandard quaternionic numerical range of $A$), is denoted by $W_\phi^{(\alpha)}(A)$. If $\alpha \neq 0$, then $W_\phi^{(\alpha)}(A)= \phi(\gamma) W_\phi^{(1)}(A) \gamma$ for some quaternion $\gamma$ with $\phi(\gamma) \gamma = \alpha$. So, the focus is on two particular nonstandard quaternionic numerical ranges $W_\phi^{(0)}(A)$ and $W_\phi^{(1)}(A)$. In this paper, a description of the intersection of $W_{\phi}^{(1)}(.)$ with a $2-$dimensional space is given, and then by using it, $W_{\phi}^{(1)}(.)$ is studied for $\phi-$hermitian quaternion matrices. After that, a class of quaternion matrices for which $W_\phi^{(0)}(.)$ and $W_\phi^{(1)}(.)$ are equal, is given. For further research, an open problem in the form of a conjecture is also given.

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Main Subjects

Journal of Advanced Mathematical Modeling
Volume 14, Issue 4 - Serial Number 36
November 2024
Pages 140-148

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History

  • Receive Date: 25 July 2024
  • Revise Date: 04 February 2025
  • Accept Date: 24 February 2025

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