The Blum–Hanson property for 𝒞(K) spaces

“…Our aim is now to investigate spaces which do not have the BH property. We will present in particular a characterization, due to Lefèvre and Matheron [14], of the compact metric spaces K which are such that C(K) has the BH property.…”

Section: Spaces Which Do Not Have the Bh Propertymentioning

confidence: 99%

The Blum-Hanson Property

Grivaux

2019

Concrete Operators

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Given a (real or complex, separable) Banach space, and a contraction T on X, we say that T has the Blum-Hanson property if whenever x, y ∈ X are such that Tnx tends weakly to y in X as n tends to infinity, the means{1 \over N}\sum\limits_{k = 1}^N {{T^{{n_k}}}x} tend to y in norm for every strictly increasing sequence (nk) k≥1 of integers. The space X itself has the Blum-Hanson property if every contraction on X has the Blum-Hanson property. We explain the ergodic-theoretic motivation for the Blum-Hanson property, prove that Hilbert spaces have the Blum-Hanson property, and then present a recent criterion of a geometric flavor, due to Lefèvre-Matheron-Primot, which allows to retrieve essentially all the known examples of spaces with the Blum-Hanson property. Lastly, following Lefèvre-Matheron, we characterize the compact metric spaces K such that the space C(K) has the Blum-Hanson property.

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The Blum–Hanson property for 𝒞(K) spaces

Abstract: We show that if K is a compact metrizable space, then the Banach space CpKq has the so-called Blum-Hanson property exactly when K has finitely many accumulation points. We also show that the space VpNq CpβNq does not have the Blum-Hanson property.

Cited by 2 publications

References 10 publications

A note on the Blum-Hanson property of some contractions on $\mathrm{L}^p$-spaces

A note on the Blum-Hanson property of some contractions on $\mathrm{L}^p$-spaces

The Blum-Hanson Property

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