Abstract. We discuss implication relations for boundedness and growth orders of Cesàro means and Abel means of discrete semigroups and continuous semigroups of linear operators. Counterexamples are constructed to show that implication relations between two Cesàro means of different orders or between Cesàro means and Abel means are in general strict, except when the space has dimension one or two.

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For a bounded linear injection C on a Banach space X and a closed linear Ž . Ž . operator A : D A ; X ª X which commutes with C we prove that 1 the Ž . Ž . Ž . Ž . operator B which commutes with C and has range contained in the range of C, A q B is also a generator.ᮊ 1997 Academic Press

We prove two ratio Tauberian theorems and deduce two generalized Tauberian theorems for functions and sequences with values in positive cones of Banach lattices. Two counter-examples are given to show that the hypotheses in the ratio Tauberian theorems are essential. Mathematics Subject Classification (2000). 40E05, 40G05; 47A35; 47D06.