Connections between the approximative and spectral properties of metric automorphisms

To each automorphism $T$ of a Lebesgue space $(X,\mu) there corresponds a~unitary operator $U_T$ in the space $L^2(X,\mu)$, defined by the formula $(U_Tf)(x)=f(Tx)$, $f\in L^2(X,\mu)$, $x\in X$. In this note we investigate the special properties of the operator $U_T$ as a~function of the rate of approximation of the automorphism $T$ by periodic transformations (for the definition of the rate of approximation of a metric automorphism see [1]).