2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/111479Let $X$ 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 unit ball $\liminf_n ρ(T^n x, K)=0$ then there exists attracting finite-dimensional subspace $L$ (for each x in X $\lim_n ρ(T^n x, L)=0$. Also we define the compact-supercyclicity. Each infinity-dimentional $X$ has no compact-supercyclic isometries. If $T$ is a supercyclic and power bounded that $T^nx$ vanishes for each $x$.5 pagesFunctional Analysis43A60; 47A16text