Timeline for Compactness in trace class operators space
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Nov 25 at 14:38 | answer | added | Onur Oktay | timeline score: 5 | |
| Nov 23 at 2:12 | comment | added | David Gao | @YemonChoi Since the OP asked whether an easy criterion exists to see if a subset is compact, I interpreted that as asking whether an equivalent condition that may be useful in practice exists. Certainly the conditions you mentioned are sufficient, trivially so, but they are definitely not necessary, no? I don’t know of an equivalent condition either, but I do think a condition equivalent to weak sequential compactness is close enough. Though it will now be up to the OP to clarify exactly what they want. | |
| Nov 23 at 0:08 | review | Close votes | |||
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| Nov 22 at 23:54 | history | edited | Yemon Choi |
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| Nov 22 at 23:51 | comment | added | Yemon Choi | @DavidGao These are all well and good, but why would one expect a given subset of the trace class operators to have these properties? I maintain that "finite subsets are compact" (or if one wants to be a bit more fancy, "convex hulls of sequences converging to zero are compact") answers the original question that was asked, just as well as the conditions that you mention | |
| Nov 22 at 22:28 | comment | added | David Gao | If you want weak sequential compactness instead, then I’m pretty sure a necessary and sufficient condition similar to tightness exists, in analogy with Prokhorov’s theorem in the classical case. Norm compactness seems hard to arrange, unless the span of your set happens to have Schur’s property, in which case weak sequential compactness and norm compactness are the same thing. | |
| Nov 22 at 22:01 | comment | added | Yemon Choi | Finite subsets are compact. Presumably you want something more than this; could you give us some indication of what you are looking for? | |
| Nov 22 at 20:10 | history | edited | lulli_ | CC BY-SA 4.0 |
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| S Nov 22 at 19:32 | review | First questions | |||
| Nov 22 at 23:01 | |||||
| S Nov 22 at 19:32 | history | asked | lulli_ | CC BY-SA 4.0 |