Let $S,T: X \rightarrow X$ be two bounded linear operators on a Banach space $X$. Let $ST=TS$.
If asc$(ST$) is finite then is asc$(T)$ finite?
For a bounded linear operator $A$ on a Banach space $X$ ascent of $A$ (denoted by asc$(A)$) is defined as the smallest positive integer $p$ such that ker$A^p=$ ker$A^{p+1}$. If no such integer exists the we set asc$(A)=\infty$