# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## The one-sided ergodic Hilbert transform in Banach spaces

### Tom 196 / 2010

Studia Mathematica 196 (2010), 251-263 MSC: Primary 47A35, 28D05, 37A05; Secondary 47B38. DOI: 10.4064/sm196-3-3

#### Streszczenie

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 $(I-T)^r\,_{|{\overline{(I-T)X}}}$.

#### Autorzy

• Guy CohenDepartment of Electrical Engineering
Ben-Gurion University
Beer Sheva 84105, Israel
e-mail
• Christophe CunyEquipe ERIM
University of New Caledonia
Nouméa, New Caledonia
e-mail
• Michael LinDepartment of Mathematics
Ben-Gurion University
Beer Sheva 84105, Israel
e-mail

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