Timeline for Reducing subspaces of unitary operators
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 7, 2023 at 14:29 | comment | added | Christian Remling | I don't think one can say much about two operators without further assumptions. There could be no common reducing subspaces, already for $2\times 2$ matrices, or there could be many. | |
| Oct 7, 2023 at 12:36 | comment | added | bm3253 | @ChristianRemling: I know that using spectral theory is overkill for the specific example of the shift. But, this is not the only case I care about. In reality, I have two noncommuting unitary operators and need to classify those subspaces which are reducing for both. I don't know much spectral theory, so I asked this simpler question to gauge whether this is the appropriate tool to use here. As for the other comment, I don't expect the only operators to commute with U to be multiplication operators. In fact, I know this isn't the case for the operators I have in mind. | |
| Oct 7, 2023 at 7:13 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title
|
| Oct 7, 2023 at 4:42 | review | Close votes | |||
| Oct 13, 2023 at 4:15 | |||||
| Oct 7, 2023 at 4:19 | comment | added | Christian Remling | That's cracking a nut with a sledgehammer. Just observe that the reducing subspaces are the ones that are invariant under $U$ and $U^*$ (or better yet, define them this way). If $U$ is multiplication by $z$ in $L^2(T,\mu)$, then, since $p(z,\overline{z})$ is dense in $C(T)$, these subspaces are exactly the spaces $L^2(A)$, $A\subseteq T$. In general, there is also the issue mentioned by Nik. | |
| Oct 7, 2023 at 2:30 | comment | added | Nik Weaver | There's a degeneracy issue you need to watch out for. For instance, if $U$ is the identity then every projection commutes with it, not just those which are multiplication operators. So you need to worry about multiplicity, but that is the only subtlety here. | |
| S Oct 7, 2023 at 1:38 | review | First questions | |||
| Oct 7, 2023 at 5:41 | |||||
| S Oct 7, 2023 at 1:38 | history | asked | bm3253 | CC BY-SA 4.0 |