Tensor Product: Closability

Question

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?

Answer 1

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.)