On the Common Spectral Properties among Bounded Linear Operators

In the context of $A, B, C$, and $D$ be bounded linear operators on Banach space $X$ satisfying the following equations:\$$\begin{aligned}(A C)^{2}=(D B)(A C),(D B)^{2}=(A C)(D B);\\B(A C) A=B(D B) A, C(A C) D=C(D B) D.\end{aligned}$$In this note we study common spectral properties for $I-AC$ (resp. $AC$) and$I-BD$ (resp. $BD$). Then we extend Cline's formula and Jacobson’s lemma for the Drazin ( resp generalized Drazin, resp generalized Drazin-Riesz) invertibility. Additionally, the study provides some others results linear operators in Banach spaces.