Almost everywhere convergence of generalized ergodic transforms for invertible power-bounded operators in $L^p$
Tom 124 / 2011
Colloquium Mathematicum 124 (2011), 61-77 MSC: Primary 47B40, 37A30. DOI: 10.4064/cm124-1-5
We show that some results of Gaposhkin about a.e. convergence of series associated to a unitary operator $U$ acting on $L^2(X,\varSigma ,\mu )$ ($\mu $ is a $\sigma $-finite measure) may be extended to the case where $U$ is an invertible power-bounded operator acting on $L^p(X,\varSigma ,\mu )$, $p>1$. The proofs make use of the spectral integration initiated by Berkson–Gillespie and, more specifically, of recent results of the author.