  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
$U\otimes U^2\otimes\dots\otimes U^{2^n}.$ 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.
