It is well-known that the sum of two surjective operators isn't (in general) a surjective operator (for example consider $A+(-A)$). When it happens that the sum of two surjective operators is still surjective? Namely, let $A,B:X\rightarrow X$ ($X$ Hilbert space) be two surjective operators, what are the properties that we can add to operators $A$, $B$ so that $A+B$ is surjective?



Sum of two surjective operators
In finite dimensions, surjective = injective, and the condition is that $B^{-1}A$ does not have $-1$ as an eigenvalue.


