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⊗ U²⊗⊗ U2ⁿ. There is an ergodic automorphism T with its symmetric tensor power Tⁿ of simple spectrum, and T⁽ⁿ⁺¹⁾ of absolutely continuous spectrum.