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 ∈ X : lim n→∞ T n x = 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

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... multiples of T . Finally, we prove that every quasi-constricted operator T such that λT 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.