Let X be a two-sided quaternionic Banach space and let \(A, B, C: X \longrightarrow X\) be bounded right linear quaternionic operators such that \(ACA=ABA\). Let q be a non-zero quaternion. In this paper, we investigate the common properties of \((AC)^{2}-2Re(q)AC+|q|^2I\) and \((BA)^{2}-2Re(q)BA+|q|^2I\) where I stands for the identity operator on X. In particular, we show that

$$\begin{aligned} \sigma ^{S}_{{\mathcal {F}}}(AC)\backslash \{0\} = \sigma ^{S}_{{\mathcal {F}}}(BA)\backslash \{0\} \end{aligned}$$

where \(\sigma ^{S}_{{\mathcal {F}}}(.)\) is a distinguished part of the spherical spectrum.

中文翻译：

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四元数设置下有界右线性算子AC和BA的共同谱性质

设X为两侧四元数 Banach 空间，并令\(A, B, C: X \longrightarrow X\)为有界右线性四元数算子，使得\(ACA=ABA\)。令q为非零四元数。在本文中，我们研究了\((AC)^{2}-2Re(q)AC+|q|^2I\)和\((BA)^{2}-2Re(q)BA+|q的共同性质|^2I\)其中I代表X上的恒等运算符。特别是，我们表明

$$\begin{对齐} \sigma ^{S}_{{\mathcal {F}}}(AC)\反斜杠 \{0\} = \sigma ^{S}_{{\mathcal {F}}} (BA)\反斜杠\{0\}\end{对齐}$$

其中\(\sigma ^{S}_{{\mathcal {F}}}(.)\)是球面光谱的一个显着部分。