Common properties of bounded linear operators $AC$ and $BA$: Spectral theory

Let $X,Y$ be Banach spaces, $A:X \longrightarrow Y$ and $B,C:Y \longrightarrow X$ be bounded linear operators satisfying operator equation $ABA=ACA$. Recently, as extensions of Jacobson's lemma, Corach, Duggal and Harte studied common properties of $AC-I$ and $BA-I$ in algebraic viewpoint and also obtained some topological analogues. In this note, we continue to investigate common properties of $AC$ and $BA$ from the viewpoint of spectral theory. In particular, we give an affirmative answer to one question posed by Corach et al. by proving that $AC - I$ has closed range if and only if $BA - I$ has closed range.