Pointwise limit theorem for a class of unbounded operators in $\mathbb L^r$-spaces
Volume 179 / 2007
Studia Mathematica 179 (2007), 49-61
MSC: Primary 47A35, 60F15; Secondary 40G10, 47B40.
DOI: 10.4064/sm179-1-5
Abstract
We distinguish a class of unbounded operators in ${{\mathbb L}}^r$, $r\geq 1$, related to the self-adjoint operators in ${{\mathbb L}}^2$. For these operators we prove a kind of individual ergodic theorem, replacing the classical Cesàro averages by Borel summability. The result is equivalent to a version of Gaposhkin's criterion for the a.e. convergence of operators. In the proof, the theory of martingales and interpolation in ${{\mathbb L}}^r$-spaces are applied.
