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Let T be a power-bounded linear operator in a real Banach space X. We study the equality (*) $(I-T)X=z\in X:su{p}_n\parallel {\sum }_{k=0}^n{T}^{k}z\parallel <\infty$. For X separable, we show that if T satisfies and is not uniformly ergodic, then $\overline{(I-T)X}$ contains an isomorphic copy of an infinite-dimensional dual Banach space. Consequently, if X is separable and does not contain isomorphic copies of infinite-dimensional dual Banach spaces, then (*) is equivalent to uniform ergodicity. As an application, sufficient conditions for uniform ergodicity of irreducible Markov chains...