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Keywords:
weighted shift operator; principal measure; distributional chaos
Summary:
During the last ten some years, many research works were devoted to the chaotic behavior of the weighted shift operator on the Köthe sequence space. In this note, a sufficient condition ensuring that the weighted shift operator $B_{w}^{n}\colon \lambda _{p}(A)\to \lambda _{p}(A)$ defined on the Köthe sequence space $\lambda _{p}(A)$ exhibits distributional $\epsilon $-chaos for any $0< \epsilon < \mathop{\rm diam} \lambda _{p}(A)$ and any $n\in \mathbb {N}$ is obtained. Under this assumption, the principal measure of $B_{w}^{n}$ is equal to 1. In particular, every Devaney chaotic shift operator exhibits distributional $\epsilon $-chaos for any $0< \epsilon < \mathop{\rm diam} \lambda _{p}(A)$.
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