Quasi-constricted linear operators on Banach spaces
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 : limn→∞ ‖Tx‖ = 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 λT is mean ergodic for all λ in the peripheral spectrum σπ(T ) of T is constricted and power bounded, and hence has a compact attractor.