Given a Banach space $X$ and a bounded linear operator $T$ on $X$. It's well known that the essential spectrum of $T$ is invariant under additive compact perturbation.

My question is about minimal hypotheses so that this result holds.

$\star$ I am imagining a condition on the measure of non-compactness associated to quotient norm.

Given a Banach space $X$ and a bounded linear operator $T$ on $X$. It's well known that the essential spectrum of $T$ is invariant under additive compact perturbation.

My question is about minimal hypotheses so that this result holds.

$\star$ I am imagining a condition on the measure of non-compactness associated to the quotient norm.