Common spectral properties of the operator products AC and BD in quaternionic setting

Abstract

Let \(A, B, C, D : X \to X\) be bounded right linear quaternionic operators acting on a two-sided quaternionic Banach space \(X\), and suppose that
$$ACD = DBD \quad \text{and} \quad DBA = ACA.$$
Let \(q\) be a non-zero quaternion. In this paper, we first present a new extension of Jacobson’s lemma in the setting of quaternionic two-sided Banach algebras. Building upon this, we analyze the operators \((AC)^{2} - 2\operatorname{Re}(q)AC + |q|^2\mathcal{I}\) and \((BD)^{2} - 2\operatorname{Re}(q)BD + |q|^2\mathcal{I}\), where \(\mathcal{I}\) denotes the identity operator on \(X\). In particular, we establish the equality
\begin{center}
$\sigma^{S}_{\star}(AC) \setminus \{0\} = \sigma^{S}_{\star}(BD) \setminus \{0\},$
\end{center}
where \(\sigma^{S}_{\star}(\cdot)\) denotes a distinguished subset of the spherical spectrum.