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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 $AB$. In particular, are there any set theoretic inclusions or everything can happen in general like: $\sigma(A) \subset \sigma(AB)$, and conversely, $\sigma(B) \subset \sigma(AB)$, and conversely?
If we know the spectra $\sigma(A)$, $\sigma(B)$ of $A$ and $B$, can we determine the spectrum of $AB$? I would appreciate any comment or reference.

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    similar question: http://math.stackexchange.com/questions/19464/spectrum-of-the-sum-of-two-commuting-matrices/19465#194652011-01-30
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    Even in one dimension, neither $\sigma(A) \subset \sigma(AB)$ nor $\sigma(B) \subset \sigma(AB)$ need hold.2011-01-30
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    Suppose I have $\sigma(AB)= \sigma(BA)$ for complex $2\times 2$ matrices. What more properties of $A$ and $B$ we can compare?2012-10-10

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