LAUSR

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.