Timeline for Given $\sigma(AB-BA) = \{0\}$, what can be said about $\sigma(A)$ and $\sigma(B)$?
Current License: CC BY-SA 4.0
4 events
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| May 16, 2023 at 9:14 | comment | added | YCor | 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 $U$. | |
| May 16, 2023 at 8:19 | comment | added | stoic-santiago | 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? | |
| May 16, 2023 at 8:04 | comment | added | YCor | 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 $A,B$. | |
| May 16, 2023 at 7:41 | history | asked | stoic-santiago | CC BY-SA 4.0 |