Quasi-constricted linear operators on Banach spaces
Volume 144 / 2001
Abstract
Let $X$ be a Banach space over $\mathbb C$. The bounded linear operator $T$ on $X$ is called quasi-constricted if the subspace $X_0:=\{ x\in X: \lim_{n\to \infty }\| T^nx\| =0\} $ is closed and has finite codimension. We show that a power bounded linear operator $T\in L(X)$ is quasi-constricted iff it has an attractor $A$ with Hausdorff measure of noncompactness $\chi _{\| \cdot \| _1}(A)< 1$ for some equivalent norm $\| \cdot \| _1$ on $X$. Moreover, we characterize the essential spectral radius of an arbitrary bounded operator $T$ by quasi-constrictedness of scalar multiples of $T$. Finally, we prove that every quasi-constricted operator $T$ such that $\overline {\lambda }T$ is mean ergodic for all $\lambda $ in the peripheral spectrum $\sigma _\pi (T)$ of $T$ is constricted and power bounded, and hence has a compact attractor.
