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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.

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    $\begingroup$ There are at least 3 definitions of "essential spectrum" which are not equivalent (though the corresponding essential spectral radius is the same). Which definition do you mean? $\endgroup$
    – Martin Väth
    Commented Aug 5, 2021 at 18:24
  • $\begingroup$ The set of all complex numbers $\lambda$ such that $\lambda-T$ is not Fredholm operator. Since you've mentioned the essential radius, that's exactly what I am looking for: the stability of the essential radius under perturbation. $\endgroup$
    – Malik Amine
    Commented Aug 5, 2021 at 18:41

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A bounded operator $K$ on $X$ satisfies $T+K$ Fredholm for each $T$ Fredholm if and only if $K$ is inessential. See P. Aiena, "Fredholm and Local Spectral Theory, with Applications to Multipliers". Kluwer, 2004.

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