The solution of the Yang-Baxter-like operator equation for rank-one operators

Abstract

Let $A\in\mathcal{B}(\mathcal{H})$ be a rank-one operator,  solutions of the Yang-Baxter-like operator equation $AXA=XAX$ on Hilbert spaces are  investigated. We derive  necessary  and sufficient conditions for an operator $X\in\mathcal{B}(\mathcal{H})$ being  a solution of the  equation. Further,  a necessary  and sufficient condition  that  the  equation has a rank-one solution is obtained for an arbitrary opeator  $A$