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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
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1 Answer 1

You can find a lot of information about the spectrum of sums of operators in the book:

  • Tosio Kato: "Perturbation Theory of Linear Operators"

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.

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www.jstor.org/stable/1997004 –  ktata.khaled May 11 '11 at 14:29
    
Thanks, more precisely I would ask these two questions: 1. Theorems 3.1 and 3.2 concern results for the essential spectra of A+B and AB for A closed and B bounded. Does this reference (or some other reference) contain results like Theorem 3.1 and 3.2 for the USUAL spectrum (\sigma(A+B), \sigma(AB)) for A closed and B bounded? 2. What exactly are the results of this paper for essential spectra of A+B and AB in the case when A and B are bounded? –  ktata.khaled May 11 '11 at 14:33
    
the reference is www.jstor.org/stable/1997004 –  ktata.khaled May 11 '11 at 14:34
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