Tensor Product: Closability _This was a real question of mine._ Given Hilbert spaces. Then closability will be inherited on tensor products: $$A,B\text{ closable}\implies A\otimes B\text{ closable}$$ For simple tensors this is clear as: $$A\otimes B(\varphi_n\otimes\psi_n)\text{ cauchy}\implies A\varphi_n,B\psi_n\text{ cauchy}$$ But what about arbitrary tensors?

_I got it meanwhile; answers are still heartly welcome!_

For densely-defined operators it holds: $$A\text{ closable}\iff A^*\text{ densely defined}$$

But one has the inclusion: $$A^*\otimes B^*\subseteq(A\otimes B)^*$$ So the result follows from denseness of the adjoint.

_(Note that this proof works only for densely defined operator.)_