For an operator $T:X\rightarrow Y$, we let $\|T\|_{e}$ denote the essential norm of $T$, that is, the distance from the compact operators, $$\|T\|_{e}:=\textrm{d}(T,\mathcal{K}(X,Y)).$$ We set $$\zeta(T):=\inf\{\frac{\|TS\|_{e}}{\|S\|_{e}}: S\in \mathcal{L}(W,X)\setminus \mathcal{K}(W,X)\},$$ where the infimum is taken over all Banach spaces $W$ and all non-compact operators $S:W\rightarrow X$.
It is easy to see that $T$ is upper semi-Fredholm whenever $\zeta(T)>0.$ That is, the class $\{T:\zeta(T)>0\}$ is a subclass of upper semi-Fredholm operators.
Question. Are there qualitative characterizations of this class of operators?