Let $H$ be a separable Hilbert space. Let $L_1$ denote the space of trace class operators on $H$ with the trace-class norm $\|\cdot\|_1$, i.e. $\|K\|_1=Tr|K|$ for all $K\in L_1$.
Are there easy criteria to see if a subset of $L_1$ is compact?
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$\begingroup$ Finite subsets are compact. Presumably you want something more than this; could you give us some indication of what you are looking for? $\endgroup$Yemon Choi– Yemon Choi2024-11-22 22:01:10 +00:00Commented Nov 22, 2024 at 22:01
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1$\begingroup$ 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. $\endgroup$David Gao– David Gao2024-11-22 22:28:07 +00:00Commented Nov 22, 2024 at 22:28
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$\begingroup$ @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 $\endgroup$Yemon Choi– Yemon Choi2024-11-22 23:51:43 +00:00Commented Nov 22, 2024 at 23:51
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$\begingroup$ @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. $\endgroup$David Gao– David Gao2024-11-23 02:12:09 +00:00Commented Nov 23, 2024 at 2:12
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1 Answer
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The space of trace class operators is nothing but $\ell^2\hat{\otimes}\ell^2$. There is a result by Grothendieck, see e.g. the snippet from [DiestelPuglisi2009].
