0
$\begingroup$

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.

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ 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
    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
    Aug 5, 2021 at 18:41

1 Answer 1

Reset to default
3
$\begingroup$

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.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.