Transformations having homogeneous spectra
Let \(T\) be an automorphism of a Lebesgue space \((X,\mu)\) and let the operator \(\widehat Tf(x):=f(Tx)\) be the operator induced by this automorphism. The main result of the paper is the statement that the map \(R:X\times X \to X\times X\) defined by the relation \(R(x,y):=(y,Tx)\) has a simple spectrum for a generic automorphism \(T\). Since the essential range of the multiplicity function of the Cartesian product \(T\times T\) is always even, the operator \(\widehat T\times\widehat T\) has a homogeneous spectrum of multiplicity two.
scientific article; zbMATH DE number 1431876