Weak multi-sensitive compactness and mean sensitivity for linear operators

Abstract

Let ( X , T ) be a linear dynamical system defined on a Banach space X and L T : K X ∗ → K X ∗ be defined by L T S = TS , where K X ∗ = span { 〈 ⋅ , y ∗ 〉 x : y ∗ ∈ X ∗ , x ∈ X } ¯ . This paper first obtains the equivalence of the (syndetically) weak multi-sensitive compactness between ( X , T ) and ( K X ∗ , L T ) . Then, it is shown that (1) there exists a non-mean-sensitive linear dynamical system ( X , T ) satisfying that ( K X ∗ , L T ) is mean sensitive; (2) There exists a non-topologically-transitive and weakly multi-sensitive compact system, which is not syndetically weakly multi-sensitive compact. Besides, it is proved that every weakly mixing linear dynamical system defined on a separable Banach space is weakly multi-sensitive compact.