On continuous embeddings of quantum Sobolev spaces into Schatten classes ${\mathfrak{H}}_{\gamma }^{s,p}(G,H)\hookrightarrow S_p(H)$

This work investigates continuous embeddings for quantum Sobolev spaces ${\mathfrak{H}}_{\gamma }^{s,p}(G,H)$ into Schatten–von Neumann classes $S_r(H)$. We try to extend the results of Lakmon and Mensah to the case where the operators belong to Schatten classes $S_p(H)$ for $p\mathrm{\ne }2$. We establish that these quantum Sobolev spaces are Banach spaces and, by employing a duality argument, we define spaces for $p>2$.