On quantum Sobolev spaces consisting of Hilbert-Schmidt operators

In the present paper, we study quantum Sobolev spaces whose elements are operators of the Hilbert-Schmidt class. We construct these Sobolev spaces from the Fourier transform for operators. Next, we obtain continuous embedding theorems. Finally, we delve into solving partial differential equations where the unknown is an operator. The results have potential applications in quantum physics, providing a new theoretical basis for relevant research.