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.][1]


 


  [1]: https://www.researchgate.net/publication/31607282_Fredholm_and_Local_Spectral_Theory_with_Applications_to_Multipliers