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

If (Tt)t⩾0(Tt)t⩾0 is a bounded C0C0-semigroup in a Banach space X   and there exists a compact subset K⊆XK⊆X such thatlim inft→∞ρ(Ttx,K)=0(∀x∈X,‖x‖⩽1), then there exists a finite-dimensional subspace L⊆XL⊆X such thatlimt→∞ρ(Ttx,L)=0(∀x∈X).If T:X→X (X   is real or complex) is supercyclic and (‖Tn‖)n(‖Tn‖)n is bounded then (Tnx)n(Tnx)n vanishes for every x∈Xx∈X.We define the “compact-supercyclicity.” If dimX=∞dimX=∞ then X has no compact-supercyclic isometries.