The one-sided ergodic Hilbert transform in Banach spaces

Guy Cohen, Christophe Cuny, and Michael Lin. "The one-sided ergodic Hilbert transform in Banach spaces." Studia Mathematica 196.3 (2010): 251-263. <http://eudml.org/doc/285703>.

@article{GuyCohen2010,
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 $lim_\{n\} ∑_\{k=1\}^\{n\} (T^\{k\}x)/k$. We prove that weak and strong convergence are equivalent, and in a reflexive space also $sup_\{n\} ||∑_\{k=1\}^\{n\} (T^\{k\}x)/k|| < ∞$ is equivalent to the convergence. We also show that $-∑_\{k=1\}^\{∞\} (T^\{k\})/k$ (which converges on (I-T)X) is precisely the infinitesimal generator of the semigroup $(I-T)^\{r\}_\{|\overline\{(I-T)X\}\}$.},
author = {Guy Cohen, Christophe Cuny, Michael Lin},
journal = {Studia Mathematica},
keywords = {ergodic Hilbert transform; operator power series; semigroup of fractional powers},
language = {eng},
number = {3},
pages = {251-263},
title = {The one-sided ergodic Hilbert transform in Banach spaces},
url = {http://eudml.org/doc/285703},
volume = {196},
year = {2010},
}

TY - JOUR
AU - Guy Cohen
AU - Christophe Cuny
AU - Michael Lin
TI - The one-sided ergodic Hilbert transform in Banach spaces
JO - Studia Mathematica
PY - 2010
VL - 196
IS - 3
SP - 251
EP - 263
AB - 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 $lim_{n} ∑_{k=1}^{n} (T^{k}x)/k$. We prove that weak and strong convergence are equivalent, and in a reflexive space also $sup_{n} ||∑_{k=1}^{n} (T^{k}x)/k|| < ∞$ is equivalent to the convergence. We also show that $-∑_{k=1}^{∞} (T^{k})/k$ (which converges on (I-T)X) is precisely the infinitesimal generator of the semigroup $(I-T)^{r}_{|\overline{(I-T)X}}$.
LA - eng
KW - ergodic Hilbert transform; operator power series; semigroup of fractional powers
UR - http://eudml.org/doc/285703
ER -