Common Spectral Properties of Bounded Right Linear Operators AC and BA in the Quaternionic Setting

Let X be a two-sided quaternionic Banach space and let \(A, B, C: X \longrightarrow X\) $A,B,C:X⟶X$ be bounded right linear quaternionic operators such that \(ACA=ABA\) $ACA=ABA$ . Let q be a non-zero quaternion. In this paper, we investigate the common properties of \((AC)^{2}-2Re(q)AC+|q|^2I\) ${(AC)}^2-2Re(q)AC+{|q|}^{2}I$ and \((BA)^{2}-2Re(q)BA+|q|^2I\) ${(BA)}^2-2Re(q)BA+{|q|}^{2}I$ where I stands for the identity operator on X. In particular, we show that \(\begin{aligned} \sigma ^{S}_{{\mathcal {F}}}(AC)\backslash \{0\} = \sigma ^{S}_{{\mathcal {F}}}(BA)\backslash \{0\} \end{aligned}\) $$\begin{array}{c}\hfill {\sigma }_{\mathcal{F}}^S(AC)\backslash \{0\}={\sigma }_{\mathcal{F}}^S(BA)\backslash \{0\}\end{array}$$ where \(\sigma ^{S}_{{\mathcal {F}}}(.)\) ${\sigma }_{\mathcal{F}}^S(.)$ is a distinguished part of the spherical spectrum.