Occasionally attracting compact sets and compact-supercyclicity

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$in unit ball$\liminf_n \rho(T^n x, K)=0$then there exists attracting finite-dimensional subspace$L$(for each x in X$\lim_n \rho(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$.