STUDIA MATHEMATICA, vol.144, no.2, pp.169-179, 2001 (SCI-Expanded)
Let X be a Banach space over C. The bounded linear operator T on X is called quasi-constricted if the subspace X-0 := {x epsilon X : lim(n --> infinity) parallel toT(n)x parallel to = 0} is closed and has finite codimension. We show that a power bounded linear operator T epsilon L(X) is quasi-constricted iff it has an attractor A with Hausdorff measure of noncompactness chi parallel to (.)parallel to (1) (A) < 1 for some equivalent norm parallel to (.)parallel to (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 <(