# Spectrum of sum of operators on Banach spaces

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.

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Inclusions don't even hold when $A, B$ are scalar multiples of the identity. In general, if $A, B$ don't commute there's no reason to expect a simple relationship between their spectra and the spectra of their sum (and you can come up with counterexamples just in the finite-dimensional case). If $A$ and $B$ are Hermitian on a finite-dimensional Hilbert space see mathoverflow.net/questions/4224/eigenvalues-of-matrix-sums (and note David Speyer's answer for the general case). – Qiaochu Yuan May 11 '11 at 5:36

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.