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Let T be a Markov operator on an L¹-space. We study conditions under which T is mean ergodic and satisfies dim Fix(T) < ∞. Among other things we prove that the sequence $\left(n^{-1}{\sum }_{k=0}^{n-1}{T}^k\right)ₙ$ converges strongly to a rank-one projection if and only if there exists a function 0 ≠ h ∈ L¹₊ which satisfies $li{m}_{n\to \infty }\left|\right|(h-n^{-1}{\sum }_{k=0}^{n-1}{T}^{k}f)₊\left|\right|=0$ for every density f. Analogous results for strongly continuous semigroups are given.

The paper introduces a notion of quasi-compact operator net on a Banach space. It is proved that quasi-compactness of a uniform Lotz-Räbiger net ${\left(T_{\lambda }\right)}_{\lambda }$ is equivalent to quasi-compactness of some operator $T_{\lambda }$. We prove that strong convergence of a quasi-compact uniform Lotz-Räbiger net implies uniform convergence to a finite-rank projection. Precompactness of operator nets is also investigated.

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...