Extensions of Jacobson’s Lemma and Cline’s formula in the Quaternionic Setting

Abstract

Let X be a two-sided Banach quaternionic space and \(A, C, B, D: X \rightarrow X\) be the right bounded linear operators satisfying operator equation set

$$\begin{aligned} A C D=D B D \;and\; D B A=A C A. \end{aligned}$$

In this paper, we generalize Jacobson’s Lemma and investigate the common properties of \((A C)^2-2 \operatorname {Re}(q) A C+|q|^2 I\) and \((B D)^2-2 \operatorname {Re}(q) B D+|q|^2 I\) where I stands for the identity operator on X and non-zero quaternion q. In particular, we show that

$$ \sigma _{\mathcal {*}}^S(A C) \backslash \{0\}=\sigma _{\mathcal {*}}^S(B D) \backslash \{0\}, $$

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