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 $C+(-A)$.
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 $B$.