An extension of the Vu–Sine theorem and compact-supercyclicity

Abstract

If (T t ) t 0 is a bounded C 0 -semigroup in a Banach space X and there exists a compact subset K ⊆ X such thatthen there exists a finite-dimensional subspace L ⊆ X such that lim t→∞ ρ(T t x, L) = 0 (∀x ∈ X).If T : X → X (X is real or complex) is supercyclic and ( T n ) n is bounded then (T n x) n vanishes for every x ∈ X.We define the "compact-supercyclicity." If dim X = ∞ then X has no compact-supercyclic isometries.

Slowly changing vectors and the asymptotic finite-dimensionality of an operator semigroup

“…For an arbitrary Banach space, this was established in [2,3]. In [4] it was proved that for the splitting X = X 0 ⊕ L, dim L < ∞, it suffices that a compact set K attract only sometimes, i.e.,…”