The convergence of Abel averages and application to ordered Banach algebras

Abstract

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.

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Acknowledgements

The authors wish to express their indebtedness to the referee, for his suggestions and valuable comments on this paper.

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Correspondence to A. Tajmouati.

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Tajmouati, A., Barki, F. The convergence of Abel averages and application to ordered Banach algebras. Positivity (2021). https://doi.org/10.1007/s11117-021-00813-w

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Keywords

  • Abel averages
  • Abel ergodic operator
  • Ergodic element
  • Pole of the resolvent
  • Ordered Banach algebra

Mathematics Subject Classification

  • 47A35
  • 47B65
  • 06F25