Poisson’s equation for mean ergodic operators

Michael Lin, Laurian Suciu

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

1 Scopus citations

Abstract

Let T be a bounded linear operator on a Banach space χ. For T power-bounded, mean ergodicity and weak mean ergodicity are equivalent, but in general, even in Hilbert spaces, this is not so. For T weakly mean ergodic, Poisson’s equation (I − T)x = y can be solved for a given y if and only if (formula presented) converges weakly. In this paper we study, for T weakly mean ergodic, the set S(T) := {y ∈ X : supn ||An(T)y|| < ∞}. We show that for T Cesàro bounded with Tn/n → 0 weakly, S(T) is closed if and only if (I −T)χ is closed, and then T is weakly mean ergodic; if ||Tn||/n → 0, then T is uniformly ergodic. As an application of our study in reflexive spaces, we supplement the Fonf-Lin-Wojtaszczyk characterization of reflexivity of Banach spaces with a basis: Such a space χ is reflexive if and only if every weakly mean ergodic T satisfies (I − T)χ = S(T), if and only if every power-bounded mean ergodic T satisfies (I − T)7 = S(T).

Original languageEnglish
Title of host publicationContemporary Mathematics
PublisherAmerican Mathematical Society
Pages141-148
Number of pages8
DOIs
StatePublished - 1 Jan 2015

Publication series

NameContemporary Mathematics
Volume636
ISSN (Print)0271-4132
ISSN (Electronic)1098-3627

ASJC Scopus subject areas

  • Mathematics (all)

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