The one-sided ergodic Hilbert transform in Banach spaces

Guy Cohen¹ ; Christophe Cuny² ; Michael Lin³

¹Department of Electrical Engineering Ben-Gurion University Beer Sheva 84105, Israel
²Equipe ERIM University of New Caledonia Nouméa, New Caledonia
³Department of Mathematics Ben-Gurion University Beer Sheva 84105, Israel

Studia Mathematica, Tome 196 (2010) no. 3, pp. 251-263

Résumé

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

Détail

DOI : 10.4064/sm196-3-3

Keywords: power bounded operator real complex banach space study convergence one sided ergodic hilbert transform lim sum frac k prove weak strong convergence equivalent reflexive space sup sum frac k infty equivalent convergence sum infty frac which converges i t precisely infinitesimal generator semigroup i t overline i t