Quasi-constricted linear operators on Banach spaces

Eduard Yu. Emel'yanov ; Manfred P. H. Wolff

Studia Mathematica, Tome 147 (2001), p. 169-179 / Harvested from The Polish Digital Mathematics Library

Résumé

Let X be a Banach space over ℂ. The bounded linear operator T on X is called quasi-constricted if the subspace $X₀: = {x ∈ X: lim_{n→ ∞} ||Tⁿx|| = 0}$ is closed and has finite codimension. We show that a power bounded linear operator T ∈ L(X) is quasi-constricted iff it has an attractor A with Hausdorff measure of noncompactness $χ_{||·||₁}(A) < 1$ for some equivalent norm ||·||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 λ̅T is mean ergodic for all λ in the peripheral spectrum $σ_{π}(T)$ of T is constricted and power bounded, and hence has a compact attractor.

Détail

Publié le : 2001-01-01

Zbl 0985.47018

EUDML-ID : urn:eudml:doc:285196