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1997 | 122 | 1 | 75-98
Tytuł artykułu

A generalization of the uniform ergodic theorem to poles of arbitrary order

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EN
Abstrakty
EN
We obtain a generalization of the uniform ergodic theorem to the sequence $(1/n^{p}) ⅀^{n-1)_{k=0} T^k$, where T is a bounded linear operator on a Banach space and p is a positive integer. Indeed, we show that uniform convergence of the sequence above, together with an additional condition which is automatically satisfied for p = 1, is equivalent to 1 being a pole of the resolvent of T plus convergence to zero of $∥T^{n}∥/n^{p}$. Furthermore, we show that the two conditions above, together, are also equivalent to 1 being a pole of order less than or equal to p of the resolvent of T, plus a certain condition ℇ(k,p), which is less restrictive than convergence to zero of $∥T^{n}∥/n^{p}$ and generalizes the condition (called condition (ℇ-k)) introduced by K. B. Laursen and M. Mbekhta in their paper [LM2] (dealing with the case p=1).
Słowa kluczowe
Czasopismo
Rocznik
Tom
122
Numer
1
Strony
75-98
Opis fizyczny
Daty
wydano
1997
otrzymano
1996-04-15
poprawiono
1996-06-10
Twórcy
Bibliografia
  • [B] L. Burlando, Uniformly p-ergodic operators and poles of the resolvent, lecture given during the Semester "Linear Operators", Warszawa, 1994.
  • [D1] N. Dunford, Spectral theory. I. Convergence to projections, Trans. Amer. Math. Soc. 54 (1943), 185-217.
  • [D2] N. Dunford, Spectral theory, Bull. Amer. Math. Soc. 49 (1943), 637-651.
  • [H] P. R. Halmos, A Hilbert Space Problem Book, Van Nostrand, 1967.
  • [K] T. Kato, Perturbation theory for nullity. deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261-322.
  • [LM1] K. B. Laursen and M. Mbekhta, Closed range multipliers and generalized inverses, Studia Math. 107 (1993), 127-135.
  • [LM2] K. B. Laursen and M. Mbekhta, Operators with finite chain length and the ergodic theorem, Proc. Amer. Math. Soc. 123 (1995), 3443-3448.
  • [La] D. C. Lay, Spectral analysis using ascent. descent. nullity and defect, Math. Ann. 184 (1970), 197-214.
  • [Li] M. Lin, On the uniform ergodic theorem, Proc. Amer Math. Soc. 43 (1974), 337-340.
  • [MO] M. Mbekhta et O. Ouahab, Opérateur s-régulier dans un espace de Banach et théorie spectrale, Acta Sci. Math. (Szeged) 59 (1994), 525-543.
  • [MZ] M. Mbekhta et J. Zemánek, Sur le théorème ergodique uniforme et le spectre, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), 1155-1158.
  • [R1] H. C. Rönnefarth, On the differences of the consecutive powers of Banach algebra elements, in: Linear Operators, Banach Center Publ. 38, Inst. Math., Polish Acad. Sci., Warszawa, to appear.
  • [R2] H. C. Rönnefarth, On properties of the powers of a bounded linear operator and their characterization by its spectrum and resolvent, thesis, Berlin, 1996.
  • [R3] H. C. Rönnefarth, A unified approach to some recent results in the uniform ergodic theory, preprint.
  • [TL] A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, 2nd ed., Wiley, 1980.
  • [W] H.-D. Wacker, Über die Verallgemeinerung eines Ergodensatzes von Dunford, Arch. Math. (Basel) 44 (1985), 539-546.
  • [Z] J. Zemánek, On the Gelfand-Hille theorems, in: Functional Analysis and Operator Theory, Banach Center Publ. 30, Inst. Math., Polish Acad. Sci., Warszawa, 1994, 369-385.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv122i1p75bwm
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