Common properties of the operator products in spectral theory

Abstract

Let $X, Y$ be Banach spaces, $A, D: X\rightarrow Y$ and $B, C: Y\rightarrow X$ be the bounded linear operators satisfying operator equation set $$\left\{ \begin{aligned} ACD=DBD~ \\ DBA=ACA. \\ \end{aligned} \right.. $$ The concept of regularity was firstly introduced by Kordula and M$\ddot{u}$ller. In this paper, we investigate the common properties of $AC$ and $BD$ in viewpoint of regularity when $A, B, C$ and $D$ all satisfy the operator equation set above.