Weak almost periodicity of L1 contractions and coboundaries of non-singular transformations

Isaac Kornfeld, Michael Lin

Research output: Contribution to journalArticlepeer-review

13 Scopus citations

Abstract

It is well known that a weakly almost periodic operator T in a Banach space is mean ergodic, and in the complex case, also λT is mean ergodic for every |λ| = 1. We prove that a positive contraction on L1 is weakly almost periodic if (and only if) it is mean ergodic. An example shows that without positivity the result is false. In order to construct a contraction T on a complex L1 such that λT is mean ergodic whenever |λ| = 1, but T is not weakly almost periodic, we prove the following: Let τ be an invertible weakly mixing non-singular transformation of a separable atomless probability space. Then there exists a complex function φ ∈ L with |φ(x)| = 1 a.e. such that for every λ ∈ ℂ with |λ| = 1 the function f ≡ 0 is the only solution of the equation f(τx) = λφ(x)f(x). Moreover, the set of such functions φ is residual in the set of all complex unimodular measurable functions (with the L1 topology).

Original languageEnglish
Pages (from-to)225-240
Number of pages16
JournalStudia Mathematica
Volume138
Issue number3
StatePublished - 1 Dec 2000

ASJC Scopus subject areas

  • Mathematics (all)

Fingerprint

Dive into the research topics of 'Weak almost periodicity of L1 contractions and coboundaries of non-singular transformations'. Together they form a unique fingerprint.

Cite this