A note on the common spectral properties for bounded linear operators

Abstract

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

11 pages