Timeline for Existence of an "Orthogonalizing" Operator
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Sep 10, 2013 at 14:48 | vote | accept | Plog | ||
| Sep 10, 2013 at 14:47 | comment | added | BS. | What about taking an orthogonal splitting $H=H_1\oplus H_2$, with $H_i$ both isomorphic to $H$, an isometry $B:H_1\to H_2$, and set $A=[0\; -B^*;B\; 0]$ ($2\times 2$ block matrix) ? | |
| Sep 10, 2013 at 14:44 | answer | added | Ian Morris | timeline score: 3 | |
| Sep 10, 2013 at 14:34 | comment | added | Plog | Ah yes, thankyou very much I had overlooked the case of spaces of odd dimension. But even so my question is really whether or not such an operator exists in infinite dimensional space. I'm starting to think this may not be as trivial as I had hoped. | |
| Sep 10, 2013 at 14:29 | comment | added | Plog | Ah yes, should have mentioned I'm working in real space. | |
| Sep 10, 2013 at 14:27 | comment | added | Ian Morris | Are you working in real or complex space? If the former, in dimension three the characteristic polynomial of $A$ is odd and therefore has a real root, which implies the existence of a one-dimensional invariant subspace. If the latter then the existence of a one-dimensional invariant subspace also holds even in dimension two. The desired property clearly cannot hold in the presence of a one-dimensional invariant subspace which does not lie in the kernel. | |
| Sep 10, 2013 at 14:06 | review | First posts | |||
| Sep 10, 2013 at 14:06 | |||||
| Sep 10, 2013 at 13:54 | history | edited | Plog | CC BY-SA 3.0 |
edited title
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| Sep 10, 2013 at 13:50 | comment | added | Plog | Yes sorry I realized this after I posted. A non-trivial kernel indeed. In fact it would be nice to know that this is possible for a unitary operator A. | |
| Sep 10, 2013 at 13:48 | comment | added | Yemon Choi | What other conditions are you putting on A? for instance, is it allowed to have non-trivial kernel? (I assume not but you should make your assumptions/conditions more clear) | |
| Sep 10, 2013 at 13:46 | history | asked | Plog | CC BY-SA 3.0 |