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Let $\mathcal H$ be a separable Hilbert space, and $\mathfrak B(\mathcal H)$ denote the algebra of bounded linear operators on $\mathcal H$. Furthermore, let $A,B \in \mathfrak B(\mathcal H)$ be two distinct operators. Given the spectrum $\sigma([A,B])$ of the commutator $[A,B] := AB-BA$, can we say anything about $\sigma(A)$ and $\sigma(B)$? In particular, I am interested in learning about $A,B$ (or their spectra) given $\sigma([A,B]) = \{0\}$.

Thanks a lot!

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    $\begingroup$ For every pair of non-empty compact subsets $W,W'\subset\mathbf{C}$ there exist commuting $A,B$ with spectra $W,W'$. Indeed choose $U,U'$ with spectra $W,W'$, then fix $t,t'\in W,W'$, and define $A=U\oplus t\mathrm{id}$, $B=t'\mathrm{id}\oplus U'$. So knowing that $\sigma([A,B])=\{0\}$ gives no information about the spectra of $A,B$. $\endgroup$
    – YCor
    Commented May 16, 2023 at 8:04
  • $\begingroup$ Thanks! Can we associate an operator with every non-empty compact subset of $\Bbb C$? The choice of $U,U'$ is not clear. On a different note, it seems $\sigma([A,B]) = \{0\}$ gives no information about $A,B$ either? $\endgroup$
    – stoic-santiago
    Commented May 16, 2023 at 8:19
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    $\begingroup$ Yes. Take a dense sequence $(t_n)$ in this compact set $U$ and take the diagonal operator with diagonal $(t_n)$. Then the spectrum is $U$. $\endgroup$
    – YCor
    Commented May 16, 2023 at 9:14

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