Operators with finite chain length and the ergodic theorem
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- by K. B. Laursen and M. Mbekhta PDF
- Proc. Amer. Math. Soc. 123 (1995), 3443-3448 Request permission
Abstract:
With a technical assumption (E-k), which is a relaxed version of the condition ${T^n}/n \to 0,n \to \infty$, where T is a bounded linear operator on a Banach space, we prove a generalized uniform ergodic theorem which shows, inter alias, the equivalence of the finite chain length condition $(X = {(I - T)^k}X \oplus \ker {(I - T)^k})$, of closedness of ${(I - T)^k}X$, and of quasi-Fredholmness of $I - T$. One consequence, still assuming (E-k), is that $I - T$ is semi-Fredholm if and only if $I - T$ is Riesz-Schauder. Other consequences are: a uniform ergodic theorem and conditions for ergodicity for certain classes of multipliers on commutative semisimple Banach algebras.References
- Pietro Aiena, Some spectral properties of multipliers on semi-prime Banach algebras, Quaestiones Math. 18 (1995), no. 1-3, 141–154. First International Conference in Abstract Algebra (Kruger Park, 1993). MR 1340474, DOI 10.1080/16073606.1995.9631791
- Nelson Dunford, Spectral theory. I. Convergence to projections, Trans. Amer. Math. Soc. 54 (1943), 185–217. MR 8642, DOI 10.1090/S0002-9947-1943-0008642-1
- Harro G. Heuser, Functional analysis, A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1982. Translated from the German by John Horváth. MR 640429
- Tosio Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261–322. MR 107819, DOI 10.1007/BF02790238
- Jean-Philippe Labrousse, Les opérateurs quasi Fredholm: une généralisation des opérateurs semi Fredholm, Rend. Circ. Mat. Palermo (2) 29 (1980), no. 2, 161–258 (French, with English summary). MR 636072, DOI 10.1007/BF02849344
- K. B. Laursen and M. Mbekhta, Closed range multipliers and generalized inverses, Studia Math. 107 (1993), no. 2, 127–135. MR 1244571
- Michael Lin, On the uniform ergodic theorem, Proc. Amer. Math. Soc. 43 (1974), 337–340. MR 417821, DOI 10.1090/S0002-9939-1974-0417821-6 M. Mbekhta, Ascente, descente et spectre essentiel quasi-Fredholm, Pub. IRMA, Lille (21), no VI, 1990.
- Mostafa Mbekhta and Jaroslav Zemánek, Sur le théorème ergodique uniforme et le spectre, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 12, 1155–1158 (French, with English and French summaries). MR 1257230 A. Ouahab, Contributions à la théorie spectrale généralisée dans les espaces de Banach, Thèse Université des Sciences et Techniques de Lille, 1991.
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 3443-3448
- MSC: Primary 47A35; Secondary 46J20, 47A53, 47B06
- DOI: https://doi.org/10.1090/S0002-9939-1995-1277123-9
- MathSciNet review: 1277123