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).
|Number of pages||16|
|State||Published - 1 Dec 2000|
ASJC Scopus subject areas
- Mathematics (all)