Spectral properties of dynamical tensor powers, and tensor factorizations of simple Lebesgue spectrum

For every $n>0$ there is a unitary operator $U$ such that the unitary operator with simple Lebesgue spectrum is isomorphic to the tensor product $U\otimes U^2\otimes \cdots \otimes U^{2^n}\mathrm{.}$ There is an ergodic automorphism $T$ with its symmetric tensor power $T^{\odot n}$ of simple spectrum, and $T^{\odot (n+1)}$ of absolutely continuous spectrum.