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

Let X, Y be Banach spaces, and B, be bounded linear operators satisfying the operator equation . Recently, as extensions of Jacobson's lemma, Corach, Duggal and Harte studied common properties of and 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 has closed range if and only if has closed range.