Let \({\mathcal {B}}(X)\) be the Banach algebra of all bounded linear operators on a Banach space X into itself. In this paper, we extend and simplify some results concerning the convergence in norm of Abel averages of an operator \(T\in {\mathcal {B}}(X)\). In particular, we show that the Abel averages of T converge in the uniform operator topology if and only if the spectral radius \(r(T)\le 1\) and the point 1 is at most a simple pole of the resolvent of T. As a consequence, we obtain a theorem on the uniform convergence of iterates of linear operators and a Gelfand–Hille type theorem. We will also show that some of the results obtained in \({\mathcal {B}}(X)\) can be extended to any Banach algebra \({\mathcal {A}}\). Finally, we will obtain results giving conditions under which a dominated positive element in an ordered Banach algebra (OBA) is Abel ergodic, given that the dominating element is Abel ergodic.

令\（{\ mathcal {B}}（X）\）为Banach空间X上所有有界线性算子的Banach代数。在本文中，我们扩展和简化了一些关于算子\（T \ in {\ mathcal {B}}（X）\）的Abel平均范数收敛的结果。特别地，我们表明，当且仅当光谱半径\（r（T）\ le 1 \）和点1最多为T的分解子的一个简单极点时，T的Abel平均才会收敛于统一算子拓扑。结果，我们得到了线性算子迭代一致收敛的一个定理和一个Gelfand-Hille型定理。我们还将显示\（{\ mathcal {B}}（X）\）中获得的一些结果可以扩展到任何Banach代数\（{\ mathcal {A}} \）。最后，我们将获得结果，给出给定条件下有序Banach代数（OBA）中占主导地位的正元素是Abel遍历，假定该主导元素是Abel遍历。