Summary: We study the concept of uniform (quasi-)\(A\)-ergodicity for \(A\)-contractions on a Hilbert space, where \(A\) is a positive operator. More precisely, we investigate the role of closedness of certain ranges in the uniformly ergodic behavior of \(A\)-contractions. We use some known results of M.Lin [Proc.Am.Math.Soc.43, 337–340 (1974; Zbl 0252.47004)], M.Mbekhta and J.Zemánek [C. R.Acad.Sci., Paris, Sér. I Math.317, No.12, 1155–1158 (1993; Zbl 0792.47006)], and S.Grabiner and J.Zemánek [J. Oper.Theory 48, No.1, 69–81 (2002; Zbl 1019.47012)], concerning the uniform convergence of the Cesàro means of an operator, to obtain similar versions for \(A\)-contractions. Thus, we continue the study of \(A\)-ergodic operators developed earlier by the author.