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1999 | 136 | 3 | 229-253
Tytuł artykułu

Les opérateurs semi-Fredholm sur des espaces de Hilbert non séparables

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Abstrakty
EN
The aim of this paper is to study the α-semi-Fredholm operators in a nonseparable Hilbert space H for all cardinals α with $ℵ_0 ≤ α ≤ dim H$. In the first part, we find the relation between $γ_α(T)$ and $c(π_α(T))$ for all $ℵ_0$-regular cardinals α, where $γ_α$ is the reduced minimum modulus of weight α, c is the reduced minimum modulus (in a C*-algebra) and $π_α$ is the canonical surjection from B(H) onto $C_α (H) = B(H)/K_α(H)$. We study the continuity points of the maps $c_α : T → c(π_α(T))$ and $γ_α : T → γ_α(T)$. In the second part, we prove some approximation results for semi-Fredholm operators. We show that all connected components of semi-Fredholm operators of at most countable index have the same topological boundary. We show that this is not true for indices strictly greater than $ℵ_0$.
Słowa kluczowe
Czasopismo
Rocznik
Tom
136
Numer
3
Strony
229-253
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-12-07
poprawiono
1999-02-08
Twórcy
  • Bâtiment de Mathématiques M2, Université des Sciences et Technologies de Lille, F-59655 Villeneuve d'Ascq Cedex, France , skhiri@gat.univ-lille1.fr
Bibliografia
  • [1] C. Apostol, The reduced minimum modulus, Michigan Math. J. 32 (1985), 279-294.
  • [2] R. H. Bouldin, The essential minimum modulus, Indiana Univ. Math. J. 30 (1981), 513-517.
  • [3] R. H. Bouldin, Closure of invertible operators on a Hilbert space, Proc. Amer. Math. Soc. 108 (1990), 721-726.
  • [4] R. H. Bouldin, Distance to invertible operators without separability, ibid. 116 (1992), 489-497.
  • [5] R. H. Bouldin, Approximating Fredholm operators on a nonseparable Hilbert space, Glasgow Math. J. 35 (1993), 167-178.
  • [6] R. H. Bouldin, Generalization of semi-Fredholm operators, Proc. Amer. Math. Soc. 123 (1995), 3757-3763.
  • [7] N. Bourbaki, Eléments de Mathématique. Livre I. Théorie des ensembles. Chapitre 3, Ensembles ordonnés cardinaux, nombres entiers, Hermann, Paris, 1967.
  • [8] L. Burlando, Distance formulas on operators whose kernel has fixed Hilbert dimension, Rend. Mat. 10 (1990), 209-238.
  • [9] L. A. Coburn and A. Lebow, Components of invertible elements in quotient algebras of operators, Trans. Amer. Math. Soc. 130 (1968), 359-365.
  • [10] J. B. Conway, A Course in Functional Analysis, 2nd ed., Springer, New York, 1990.
  • [11] G. Edgar, D. Ernest and S. G. Lee, Weighing operator spectra, Indiana Univ. Math. J. 21 (1971), 61-80.
  • [12] D. Ernest, Operators with α -closed range, Tôhoku Math. J. 24 (1972), 45-49.
  • [13] P. A. Fillmore, J. G. Stampfli and J. P. Williams, On the essential numerical range, the essential spectrum, and a problem of Halmos, Acta Sci. Math. (Szeged) 33 (1972), 179-192.
  • [13] S. Goldberg, Unbounded Linear Operators, McGraw-Hill, New York, 1966.
  • [14] B. Gramsch, Eine idealstruktur Banachscher Operatoralgebren, J. Reine Angew. Math. 225 (1967), 97-115.
  • [15] P. R. Halmos, A Hilbert Space Problem Book, D. Van Nostrand, 1967.
  • [16] R. E. Harte and M. Mbekhta, On generalized inverses in C*-algebras, Studia Math. 103 (1992), 71-77.
  • [17] R. E. Harte and M. Mbekhta, Generalized inverses in C*-algebras II, ibid. 106 (1993), 129-138.
  • [18] D. A. Herrero, Approximation of Hilbert Space Operators, Vol. I, Pitman, Boston, 1982.
  • [19] S. Izumino and Y. Kato, The closure of invertible operators on a Hilbert space, Acta Sci. Math. (Szeged) 49 (1985), 321-327.
  • [20] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1966.
  • [21] T. Kato, Perturbation theory for nullity, deficiency, and other quantities of linear operators, J. Anal. Math. 6 (1958), 261-322.
  • [22] E. Luft, The two-sided closed ideals of the algebra of bounded linear operators of Hilbert space, Czechoslovak Math. J. 18 (1968), 595-605.
  • [23] M. Mbekhta, Conorme et inverse généralisé dans les C*-algèbres, Canad. Math. Bull. 35 (1992), 515-522.
  • [24] H. Skhiri, On the topological boundary of semi-Fredholm operators, Proc. Amer. Math. Soc. 126 (1998), 1381-1389.
  • [25] H. Skhiri, Opérateurs semi-Fredholm : structures et approximations, Thèse de doctorat, Université de Lille 1, 1997.
  • [26] A. Ströh, Regular liftings in C*-algebras, Bull. Polish Acad. Sci. Math. 42 (1994), 1-7.
  • [27] P. Y. Wu, Approximation by invertible and noninvertible operators, J. Approx. Theory 56 (1989), 267-276.
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