  Let $X$ and $Y$ be Banach spaces, $A\,:\,X\rightarrow Y$ and
$B,\,C\,:\,Y\rightarrow X$ be bounded linear operators. We prove that if
$A(BA)^2=ABACA=ACABA=(AC)^2A,$ then
$$\sigma_{*}(AC)\setminus\{0\}=\sigma_{*}(BA)\setminus\{0\}$$ where $\sigma_*$
runs over a large of spectra originated by regularities.
