Tauberian ergodic theorem for normal contraction operators
It follows from the paper by \textit{D. L. Burkholder} and \textit{Y. S. Chow} [Proc. Am. Math. Soc. 12, 490-495 (1961; Zbl 0106.332)] that if T is a positive (in the Hilbert space sense) and selfadjoint contraction in \(L_ 2=L_ 2(\mu)\), then \[ (*)\quad \lim_{n\to \infty}T^ nf-n^{- 1}\sum^{n-1}_{k=0}T^ kf)=0\quad a.e.\quad for\quad all\quad f\in L_ 2. \] In the paper under review the author generalizes this result by proving that (*) holds for normal contractions T in \(L_ 2\) whose spectrum is contained in some closed Stolz region in the unit disk with vertex at \(z=1\). The paper contains also a class of examples indicating that the last condition on the spectrum cannot be essentially weakened.
scientific article; zbMATH DE number 3895692