Occasionally attracting compact sets and compact-supercyclicity

K. Storozhuk

paper

Occasionally attracting compact sets and compact-supercyclicity

arXiv:math/0604159

Abstract

Let $X$ be a real or complex Banach space and $T_{t} : X \to X$ is a power bounded operator (or a $C_{0}$ -semigroup). If there exists a "occasionally" attracting compact subset K (for each x $in u ni t ba l l$ \liminf_n ρ(T^n x, K)=0 $t h e n t h er ee x i s t s a tt r a c t in g f ini t e - d im e n s i o na l s u b s p a ce$ L $(f or e a c h x in X$ \lim_n ρ(T^n x, L)=0 $. A l so w e d e f in e t h eco m p a c t - s u p er cy c l i c i t y . E a c hin f ini t y - d im e n t i o na l$ X $ha s n oco m p a c t - s u p er cy c l i c i so m e t r i es . I f$ T $i s a s u p er cy c l i c an d p o w er b o u n d e d t ha t$ T^nx $v ani s h es f or e a c h$ x$.

5 pages