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1999 | 134 | 2 | 143-151
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A resolvent condition implying power boundedness

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The Ritt and Kreiss resolvent conditions are related to the behaviour of the powers and their various means. In particular, it is shown that the Ritt condition implies the power boundedness. This improves the Nevanlinna characterization of the sublinear decay of the differences of the consecutive powers in the Esterle-Katznelson-Tzafriri theorem, and actually characterizes the analytic Ritt condition by two geometric properties of the powers.
Słowa kluczowe
Czasopismo
Rocznik
Tom
134
Numer
2
Strony
143-151
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-02-15
poprawiono
1998-11-05
Twórcy
autor
  • Mathematical Institute, Technical University of Budapest, Egry József u. 2, H. ép., II. em., H-1521 Budapest XI, Hungary, bnagy@math.bme.hu
  • Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, P.O. Box 137, 00-950 Warszawa, Poland, zemanek@impan.gov.pl
Bibliografia
  • [A1] G. R. Allan, Sums of idempotents and a lemma of N. J. Kalton, Studia Math. 121 (1996), 185-192.
  • [A2] G. R. Allan, Power-bounded elements and radical Banach algebras, in: Linear Operators, J. Janas, F. H. Szafraniec and J. Zemánek (eds.), Banach Center Publ. 38, Inst. Math., Polish Acad. Sci., 1997, 9-16.
  • [ARa] G. R. Allan and T. J. Ransford, Power-dominated elements in a Banach algebra, Studia Math. 94 (1989), 63-79.
  • [D] N. Dunford, Spectral theory. I. Convergence to projections, Trans. Amer. Math. Soc. 54 (1943), 185-217.
  • [EHP] P. Erdős, F. Herzog and G. Piranian, On Taylor series of functions regular in Gaier regions, Arch. Math. (Basel) 5 (1954), 39-52.
  • [Es] J. Esterle, Quasimultipliers, representations of $H^∞$, and the closed ideal problem for commutative Banach algebras, in: Radical Banach Algebras and Automatic Continuity (Long Beach, Calif., 1981), J. M. Bachar, W. G. Bade, P. C. Curtis Jr., H. G. Dales, and M. P. Thomas (eds.), Lecture Notes in Math. 975, Springer, 1983, 66-162.
  • [F] H. O. Fattorini, The Cauchy Problem, Addison-Wesley, Reading, Mass., 1983.
  • [Go] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Univ. Press, New York, 1985.
  • [GHu] J. J. Grobler and C. B. Huijsmans, Doubly Abel bounded operators with single spectrum, Quaestiones Math. 18 (1995), 397-406.
  • [Ha] P. R. Halmos, A Hilbert Space Problem Book, Van Nostrand, Princeton, 1967.
  • [Hi] E. Hille, Remarks on ergodic theorems, Trans. Amer. Math. Soc. 57 (1945), 246-269.
  • [KT] Y. Katznelson and L. Tzafriri, On power bounded operators, J. Funct. Anal. 68 (1986), 313-328.
  • [Ki] J. Kisyński, On resolvents and semigroups associated with the Dirichlet problem for an elliptic differential operator of second order with a Lévy perturbation, Wydawnictwa Uczelniane Politechniki Lubelskiej, Politechnika Lubelska, Lublin, 1998.
  • [L] Yu. Lyubich, Spectral localization, power boundedness and invariant subspaces under Ritt's type condition, Studia Math. 134 (1999), 153-167.
  • [LZ] Yu. Lyubich and J. Zemánek, Precompactness in the uniform ergodic theory, ibid. 112 (1994), 89-97.
  • [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.
  • [N1] O. Nevanlinna, Convergence of Iterations for Linear Equations, Birkhäuser, Basel, 1993.
  • [N2] O. Nevanlinna, On the growth of the resolvent operators for power bounded operators, in: Linear Operators, J. Janas, F. H. Szafraniec and J. Zemánek (eds.), Banach Center Publ. 38, Inst. Math., Polish Acad. Sci., 1997, 247-264.
  • [Pa] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
  • [PóS] G. Pólya und G. Szegő, Aufgaben und Lehrsätze aus der Analysis, Springer, Berlin, 1964.
  • [Py] T. Pytlik, Analytic semigroups in Banach algebras and a theorem of Hille, Colloq. Math. 51 (1987), 287-294.
  • [R] R. K. Ritt, A condition that $lim_n→∞ n^-1T^n=0$, Proc. Amer. Math. Soc. 4 (1953), 898-899.
  • [Rö1] H. C. Rönnefarth, On properties of the powers of a bounded linear operator and their characterization by its spectrum and resolvent, Dissertation, Technische Universität Berlin, Berlin, 1996.
  • [Rö2] H. C. Rönnefarth, On the differences of the consecutive powers of Banach algebra elements, in: Linear Operators, J. Janas, F. H. Szafraniec and J. Zemánek (eds.), Banach Center Publ. 38, Inst. Math., Polish Acad. Sci., 1997, 297-314.
  • [Sh] A. L. Shields, On Möbius bounded operators, Acta Sci. Math. (Szeged) 40 (1978), 371-374.
  • [StW] J. C. Strikwerda and B. A. Wade, A survey of the Kreiss matrix theorem for power bounded families of matrices and its extensions, in: Linear Operators, J. Janas, F. H. Szafraniec and J. Zemánek (eds.), Banach Center Publ. 38, Inst. Math., Polish Acad. Sci., 1997, 339-360.
  • [Św] A. Święch, A note on the differences of the consecutive powers of operators, ibid., 381-383.
  • [Tad] E. Tadmor, The resolvent condition and uniform power-boundedness, Linear Algebra Appl. 80 (1986), 250-252.
  • [TaY] K. Tanahashi and S. Yamagami, Spectral inclusion relations for T, T|Y, and T/Y, Proc. Amer. Math. Soc. 116 (1992), 763-768.
  • [Tay] A. E. Taylor, Introduction to Functional Analysis, Wiley, New York, 1958.
  • [V] G. Valiron, Fonctions Analytiques, Presses Universitaires de France, Paris, 1954.
  • [Z1] J. Zemánek, On the Gelfand-Hille theorems, in: Functional Analysis and Operator Theory, J. Zemánek (ed.), Banach Center Publ. 30, Inst. Math., Polish Acad. Sci., 1994, 369-385.
  • [Z2] J. Zemánek, Problem, in: Banach Algebras '97, Proc. 13th Internat. Conf. on Banach Algebras (Blaubeuren, 1997), E. Albrecht and M. Mathieu (eds.), de Gruyter, Berlin, 1998, 560.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv134i2p143bwm
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