The one-sided ergodic Hilbert transform of normal contractions

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

7 Scopus citations

Abstract

Let T be a normal contraction on a Hilbert space H. For f ∈ H we study the one-sided ergodic Hilbert transform (Formula Presented). We prove that weak and strong convergence are equivalent, and show that the convergence is equivalent to convergence of the series (Formula Presented). When (Formula Presented), the transform is shown to be precisely minus the infinitesimal generator of the strongly continuous semi-group {(I − T)r}r≥0. The equivalence of weak and strong convergence of the transform is proved also for T an isometry or the dual of an isometry. For a general contraction T, we obtain that convergence of the series (Formula Presented) implies strong convergence of (Formula Presented).

Original languageEnglish
Title of host publicationCharacteristic Functions, Scattering Functions and Transfer Functions
EditorsDaniel Alpay, Victor Vinnikov
PublisherSpringer International Publishing
Pages77-98
Number of pages22
ISBN (Print)9783034601825
DOIs
StatePublished - 1 Jan 2010
EventInternational Conference on Characteristic functions and transfer functions in operator theory and system theory, 2007 - Beersheba, Israel
Duration: 9 Jul 200713 Jul 2007

Publication series

NameOperator Theory: Advances and Applications
Volume197
ISSN (Print)0255-0156
ISSN (Electronic)2296-4878

Conference

ConferenceInternational Conference on Characteristic functions and transfer functions in operator theory and system theory, 2007
Country/TerritoryIsrael
CityBeersheba
Period9/07/0713/07/07

Keywords

  • Normal contractions
  • One-sided ergodic Hilbert transform

ASJC Scopus subject areas

  • Analysis

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