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

TL;DR

This paper explores the spectral properties of tensor powers of unitary operators and ergodic automorphisms, revealing new insights into their spectrum types and tensor factorizations.

Contribution

It demonstrates the existence of unitary operators with specific spectral tensor product structures and constructs ergodic automorphisms with distinct spectral behaviors in their tensor powers.

Findings

Existence of unitary operators with tensor products isomorphic to simple Lebesgue spectrum

Construction of ergodic automorphisms with simple spectrum in symmetric tensor powers

Identification of spectral transition from simple to absolutely continuous in tensor powers

Abstract

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.