Let X be a Banach space over ℂ. The bounded linear operator T on X is called quasi-constricted if the subspace $X₀:=x\in X:li{m}_{n\to \infty }\left|\right|Tⁿx\left|\right|=0$ 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 ${\chi }_{\left|\right|·\left|\right|₁}\left(A\right)<1$ 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...