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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References
Alekhno, E.A.: The irreducibility in ordered Banach algebras. Positivity 16, 143–176 (2012)
Badiozzaman, A.J., Thorpe, B.: A uniform Ergodic theorem and summability. Bull. Lond. Math. Soc. 24, 351–360 (1992)
Behrendt, D., Raubenheimer, H.: On domination of inessential elements in ordered Banach algebras. Ill. J. Math. 51(3), 927–936 (2007)
Benjamin, R., Mouton, S.: Fredholm theory in ordered Banach algebras. Quaest. Math. 39(5), 643–664 (2016)
Burlando, L.: A generalization of the uniformly Ergodic theorem to poles of arbitrary order. Stud. Math. 122, 75–98 (1997)
Braatvedt, G., Brits, R., Raubenheimer, H.: Gelfand–Hille type theorems in ordered Banach algebras. Positivity 13, 39–50 (2009)
Chen, J.C., Shaw, S.Y.: Growth order and stability of discrete semigroups. Nonlinear Anal. 71, 2879–2882 (2009)
Eberlein, W.F.: Abstract ergodic theorems and weak almost periodic functions. Trans. Amer. Math. Soc. 67, 217–240 (1949)
Hille, E.: Remarks on ergodic theorems. Trans. Am. Math. Soc. 57, 246–269 (1945)
Jachymski, J.: Convergence of iterates of linear operators and the Kelisky–Rivlin type theorems. Stud. Math. 195, 99–113 (2009)
Krengel, U.: Ergodic Theorems, Walter de Gruyter Studies in Mathematics 6. Walter de Gruyter, Berlin (1985)
Lin, M.: On the uniform ergodic theorem. Proc. Am. Math. Soc. 43, 337–340 (1974)
Lin, M.: On the uniform ergodic theorem II. Proc. Am. Math. Soc. 46, 217–225 (1974)
Lin, M., Shoikhet, D., Suciu, L.: Remarks on uniform Ergodic theorems. Acta Sci. Math. (Szeged) 81, 251–283 (2015)
Mbekhta, M., Zemànek, J.: Sur le théorème ergodique uniforme et le spectre. C. R. Acad. Sci. Paris série I Math. 317, 1155–1158 (1993)
Mouton, S.: Convergence properties of positive elements in Banach algebras. Math. Proc. R. Ir. Acad. Sect. A 102, 149–162 (2002)
Mouton, S., Muzundu, K.: Domination by ergodic elements in ordered Banach algebras. Positivity 18, 119–130 (2014)
Mouton, S., Raubenheimer, H.: More spectral theory in ordered Banach algebras. Positivity 1, 305–317 (1997)
Mouton, S., Raubenheimer, H.: Spectral theory in ordered Banach algebras. Positivity 21, 755–786 (2017)
Taylor, A.E., Lay, D.C.: Introduction to Functional Analysis. Wiley, New York (1980)
Wickstead, A.W.: Ordered Banach algebras and multi-norms: some open problems. Positivity 21, 817–823 (2017)
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The authors wish to express their indebtedness to the referee, for his suggestions and valuable comments on this paper.
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Tajmouati, A., Barki, F. The convergence of Abel averages and application to ordered Banach algebras. Positivity 25, 1255–1266 (2021). https://doi.org/10.1007/s11117-021-00813-w
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DOI: https://doi.org/10.1007/s11117-021-00813-w