0
$\begingroup$

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 $C+(-A)$.

$\endgroup$
  • 2
    $\begingroup$ 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
1
$\begingroup$

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 $B$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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