The one-sided ergodic Hilbert transform in Banach spaces
Volume 196 / 2010
Studia Mathematica 196 (2010), 251-263
MSC: Primary 47A35, 28D05, 37A05; Secondary 47B38.
DOI: 10.4064/sm196-3-3
Abstract
Let $T$ be a power-bounded operator on a (real or complex) Banach space. We study the convergence of the one-sided ergodic Hilbert transform $\lim_n \sum_{k=1}^n \frac{T^k x}k$. We prove that weak and strong convergence are equivalent, and in a reflexive space also $\sup_n \|\sum_{k=1}^n \frac{T^k x}k\| < \infty$ is equivalent to the convergence. We also show that $-\sum_{k=1}^\infty \frac{T^k}k$ (which converges on $(I-T)X$) is precisely the infinitesimal generator of the semigroup $(I-T)^r\,_{|{\overline{(I-T)X}}}$.
