Suppose it is known that $\scrM$ is an invariant subspace for $A$. What invariant subspaces for $A\otimes A$ can be obtained from this information alone?

Solution. It is $\scrM\otimes \scrM$ that is an invariant subspace of $A\otimes A$. Indeed, if $x,y\in M$, then $$\bex (A\otimes A)(x\otimes y)=(Ax)\otimes (Ay)\in M\otimes M. \eex$$

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.4.5