A New Perspective on Eigenvalues and Eigenvectors of Bicomplex Linear Operators
Anjali Anjali, Akhil Prakash, Amita, Prabhat Kumar
TL;DR
The paper studies the spectral theory of bicomplex linear operators on $\mathbb{C}_2^n$ by exploiting the idempotent decomposition $T=e_1T_1+e_2T_2$, linking eigenvalues in $\mathbb{C}_1$ with modified eigenvalues in $\mathbb{C}_2$. It establishes fundamental relations between eigenvalues of the component operators $T_1$ and $T_2$ and the (modified) eigenstructure of $T$, and introduces structured notions of eigenspaces and modified eigenspaces with precise Cartesian-type decompositions. Key results include the inclusion $\Upsilon_1\times_e\Upsilon_2\subset\Upsilon$, the characterization that an eigenvalue $\lambda$ of $T$ lies in $\Upsilon_1\cup\Upsilon_2$, and the detailed descriptions of $^T(ME)_{\kappa}$ under various cases of $\kappa_1,\kappa_2$. The work also proves the existence of infinitely many modified eigenvalues arising from eigenvalues of $T_1$ or $T_2$, and proposes an open problem regarding the sum vs direct sum of modified eigenspaces, contributing a rigorous framework for bicomplex spectral theory with potential applications to bicomplex matrix analysis.
Abstract
This paper deals with eigenvalues and eigenvectors of bicomplex linear operators defined on bicomplex space. We investigate the properties of these operators in the context of eigenvalues and eigenvectors, along with some relevant theorems. Several theorems are explored to establish conditions for bicomplex eigenvalues and eigenvectors. Additionally, we examine the structure of the eigenspaces corresponding to these eigenvalues,analyze their properties, and discuss relevant theorems.