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Weak almost periodicity of $L_1$ contractions and coboundaries of non-singular transformations

100%

Kornfeld I., Lin M.

Studia Mathematica

2000

tom 138

nr 3

225-240

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 $L_1$ 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 $L_1$ 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 ⨍ ≡ 0 is the only solution of the equation ⨍(τx)=λφ(x)⨍(x). Moreover, the set of such functions φ is residual in the set of all complex unimodular measurable functions (with the $L_1$ topology)

Positive L¹ operators associated with nonsingular mappings and an example of E. Hille

100%

Kornfeld I., Kosek W.

Colloquium Mathematicum

2003

tom 98

nr 1

63-77

E. Hille [Hi1] gave an example of an operator in L¹[0,1] satisfying the mean ergodic theorem (MET) and such that supₙ||Tⁿ|| = ∞ (actually, $||Tⁿ|| ∼ n^{1/4}$). This was the first example of a non-power bounded mean ergodic L¹ operator. In this note, the possible rates of growth (in n) of the norms of Tⁿ for such operators are studied. We show that, for every γ > 0, there are positive L¹ operators T satisfying the MET with $lim_{n→ ∞} ||Tⁿ||/n^{1-γ} = ∞. In the class of positive operators these examples are the best possible in the sense that for every such operator T there exists a γ₀ > 0 such that $lim sup_{n→ ∞} ||Tⁿ||/n^{1-γ₀} = 0$. A class of numerical sequences αₙ, intimately related to the problem of the growth of norms, is introduced, and it is shown that for every sequence αₙ in this class one can get ||Tⁿ|| ≥ αₙ (n = 1,2,...) for some T. Our examples can be realized in a class of positive L¹ operators associated with piecewise linear mappings of [0,1].

Ograniczanie wyników

1 Colloquium Mathematicum

1 Studia Mathematica

2 Kornfeld I.

1 Kosek W.

1 Lin M.

1 2003

1 2000

Wyniki wyszukiwania

Weak almost periodicity of $L_1$ contractions and coboundaries of non-singular transformations

Positive L¹ operators associated with nonsingular mappings and an example of E. Hille