# If $A+B$ is a compact non-zero operator, which properties follow for $A$ and $B$?

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 $$C+(-A)$$.

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