There is a classical theorem of Riesz-Kolmogorov that characterizes compact embedding in $L^p$-spaces of some subspace of them. A generalization to arbitrary metric spaces has been recently obtained by Hanche-Olsen and Holden - their criterion can be in particular used to discuss compactness of the embedding of some subspace of $\ell^p$.

Is there anything similar about trace-class embeddings? There are some results of Rellich-Khondrakhov-type yielding trace-class embedding of Sobolev spaces $H^k$ into $L^2$ (they are due to Gramsch and Maurin, later elaborated on in Adams' book) (spoiler: the embedding of $H^1(0,1)$ in $L^2(0,1)$ is of trace class), but they are formulated for domains and I do not see how to generalize them to general measure spaces.

Even the case of subspaces of $\ell^2(\mathbb N)$ would be rather interesting for me.