Quasi-constricted linear operators on Banach spaces

Eduard Yu. Emel'yanov¹ ; Manfred P. H. Wolff²

¹ Sobolev Institute of Mathematics at Novosibirsk Acad. Koptyug pr. 4 630090 Novosibirsk, Russia Current address Mathematisches Institut der Universität Tübingen A. D. Morgenstelle 2 D-72076 Tübingen, Germany
² Mathematisches Institut der Universität Tübingen A. D. Morgenstelle 2 D-72076 Tübingen, Germany

Studia Mathematica, Tome 144 (2001) no. 2, pp. 169-179

Résumé

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.

Détail

DOI : 10.4064/sm144-2-5

Keywords: banach space mathbb bounded linear operator called quasi constricted subspace lim infty closed has finite codimension power bounded linear operator quasi constricted has attractor hausdorff measure noncompactness chi cdot equivalent norm cdot moreover characterize essential spectral radius arbitrary bounded operator quasi constrictedness scalar multiples finally prove every quasi constricted operator overline lambda mean ergodic lambda peripheral spectrum sigma constricted power bounded hence has compact attractor