Assume we are in $l^p$ spaces or at least Banach spaces. I'm trying to find out what the knowledge that the linear operator $A+B$ is compact tells me about the properties of the linear operators $A$ and $B$. Does it say anything about boundedness or compactness of one or both of them, for example.

Basically my question is whether or not $A+B$ being compact tells me that at least one or both of $A$ and $B$ are compact. Now if we allow $A+B=0$ there are obvious counter examples such as just taking $I-I$ but I have not been able to come up with counter examples for $A+B \neq 0$ where also $A \neq 0$ and $B \neq 0$. So everything being non-zero, if I write a compact operator as the sum of two operators, do one or both of them also have to be compact? Are there known results on this?

I know that if $A+B$ is compact and $A$ is also taken to be compact, then this implies compactness of $B$. That is because since $C=A+B$ is compact and $A$ is too then we can write $B$ as the sum of two compact operators as $C+(-A)$.

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    $\begingroup$ What about this: Fix $K$ a compact operator and let $A$ be any operator you want, then define $B:=K-A$. $\endgroup$ – Jose27 Mar 29 at 5:39
  • $\begingroup$ I was editing my question to account for a similar realization just as you posted the comment but thank you very much either way! $\endgroup$ – MM8 Mar 29 at 5:42
  • $\begingroup$ Taking $B=-A$ makes it clear that your guesses are all false. $\endgroup$ – Kavi Rama Murthy Mar 29 at 5:42
  • $\begingroup$ You cannot take $B=-A$ as I explicitly asked for $A+B \neq 0$. $\endgroup$ – MM8 Mar 29 at 5:42

If $K$ is compact then $I-K$ cannot be compact. Otherwise $I=I-K+K$ would be compact. Now set $A=K-I$ and $B=K+I$, then $A+B$ is compact but neither is $A$ nor $B$.


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