Let $A$ and $B$ be two operators on a Banach space $X$. I am interested in the relationship between the spectra of $A$, $B$ and $A+B$. In particular, are there any set theoretic inclusions or everything can happen in general like: $\sigma(A)\subset\sigma(A+B)$, and conversely, $\sigma(B)\subset\sigma(A+B)$, and conversely? If we know the spectra $\sigma(A)$, $\sigma(B)$ of $A$ and $B$, can we determine the spectrum of $A+B$? I would appreciate any comment or reference.
You can find a lot of information about the spectrum of sums of operators in the book:
Chapter Four "Stability theorems", paragraph 3 "Perturbation of the spectrum" (that's about linear operators on infinite dimensional Banach spaces, other cases are treated as well in the book).
As shown there, it is possible to prove some theorems about the spectrum of $A + B$, where A is a known operator (you know something about its spectrum) and B is a perturbation of A in the sense that it is small compared to A in a certain way, which can be made precise in different ways with different uses.
These theorems are the best results that I know which show that what happens to the spectrum of the sum is not completey arbitrary in certain settings.