Weighted ergodic theorems for weakly mixing transformation groups
Let \((T_ t)\) denote the group of unitary operators on \(L^ 2(\Omega,F,P)\) corresponding to a one parameter group of measure- preserving transformations on the probability space \((\Omega,F,P)\). W is the class of all bounded complex valued functions p on R such that \(m(p)=\lim_{r\to \infty}\int^{r}_{0}p(t)dt\) exists. For \(p\in W\), \(f\in L^ 2(\Omega)\) write \(A_ r(f,p)(x)=(1/r)\int^{r}_{0}p(t)T_ tf(x)dt.\) The following theorems are proved: Theorem 1. The following conditions are equivalent, (a) \((T_ t)\) is weakly mixing, \((b)\quad \forall p\in W,\quad A_ r(f,p)\) converges in the mean to \(m(p)E(f),\quad \forall f\in L^ 2(\Omega).\) \((E(f)=\int_{\Omega}f dP).\) Theorem 2. \((T_ t)\) is weakly mixing if and only if for any periodic function \(f\in W,\) \(A_ r(f,p)\) converges a.e. to \(m(p)E(f)\forall f\in L^ 2(\Omega)\).
one parameter group of measure-preserving transformations
scientific article; zbMATH DE number 4200535