Analytic vectors of tensor product operators

Question

For every $k=1,\ldots,r$, let $A_k$ be a (not necessarily bounded) self-adjoint operator on a Hilbert space $\mathcal{H}_k$, and let $\psi_k$ be an analytic vector for $A_k$. Is it true that $\psi_1\otimes\ldots\otimes\psi_r$ is an analytic vector for $A_1\otimes\ldots\otimes A_r$?