Operational quantities derived from the norm and generalized Fredholm theory

Manuel Gonzalez; Antonio Martinón

Commentationes Mathematicae Universitatis Carolinae (1991)

  • Volume: 32, Issue: 4, page 645-657
  • ISSN: 0010-2628

Abstract

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We introduce and study some operational quantities associated to a space ideal 𝔸 . These quantities are used to define generalized semi-Fredholm operators associated to 𝔸 , and the corresponding perturbation classes which extend the strictly singular and strictly cosingular operators, and we study the generalized Fredholm theory obtained in this way. Finally we present some examples and show that the classes of generalized semi-Fredholm operators are non-trivial for several classical space ideals.

How to cite

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Gonzalez, Manuel, and Martinón, Antonio. "Operational quantities derived from the norm and generalized Fredholm theory." Commentationes Mathematicae Universitatis Carolinae 32.4 (1991): 645-657. <http://eudml.org/doc/247305>.

@article{Gonzalez1991,
abstract = {We introduce and study some operational quantities associated to a space ideal $\mathbb \{A\}$. These quantities are used to define generalized semi-Fredholm operators associated to $\mathbb \{A\}$, and the corresponding perturbation classes which extend the strictly singular and strictly cosingular operators, and we study the generalized Fredholm theory obtained in this way. Finally we present some examples and show that the classes of generalized semi-Fredholm operators are non-trivial for several classical space ideals.},
author = {Gonzalez, Manuel, Martinón, Antonio},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {semi-Fredholm operator; strictly singular operator; perturbation; upper and lower semi-Fredholm operators; compact operators; strictly singular and strictly cosingular operators; operational quantity; space ideal},
language = {eng},
number = {4},
pages = {645-657},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Operational quantities derived from the norm and generalized Fredholm theory},
url = {http://eudml.org/doc/247305},
volume = {32},
year = {1991},
}

TY - JOUR
AU - Gonzalez, Manuel
AU - Martinón, Antonio
TI - Operational quantities derived from the norm and generalized Fredholm theory
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1991
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 32
IS - 4
SP - 645
EP - 657
AB - We introduce and study some operational quantities associated to a space ideal $\mathbb {A}$. These quantities are used to define generalized semi-Fredholm operators associated to $\mathbb {A}$, and the corresponding perturbation classes which extend the strictly singular and strictly cosingular operators, and we study the generalized Fredholm theory obtained in this way. Finally we present some examples and show that the classes of generalized semi-Fredholm operators are non-trivial for several classical space ideals.
LA - eng
KW - semi-Fredholm operator; strictly singular operator; perturbation; upper and lower semi-Fredholm operators; compact operators; strictly singular and strictly cosingular operators; operational quantity; space ideal
UR - http://eudml.org/doc/247305
ER -

References

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  2. Alvarez T., Gonzalez M., Some examples of tauberian operators, Proc. Amer. Math. Soc. 111 (1991), 1023-1027. (1991) MR1033955
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  4. Alvarez T., Gonzalez M., Onieva V.M., Characterizing two classes of operator ideals, Contribuciones Matematicas Homenaje Prof. Plans, Univ. Zaragoza (1990), 7-21. 
  5. Fajnshtejn A.S., On measures of noncompactness of linear operators and analogs of the minimal modulus for semi-Fredholm operators (in Russian), Spektr. Teor. Oper. 6 (1985), 182-185, Zbl. 634#47010. (1985) 
  6. Goldberg S., Unbounded Linear Operators, McGraw Hill, 1966. Zbl1152.47001MR0200692
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  8. Gonzalez M., Martinon A., A generalization of semi-Fredholm operators, preprint. MR1285738
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  10. Gonzalez M., Onieva V.M., Characterizations of tauberian operators, Proc. Amer. Math. Soc. 108 (1990), 399-405. (1990) Zbl0704.47016MR0994777
  11. Kadets M.I., Note on the gap between subspaces, Funct. Anal. Appl. 9 (1975), 156-157. (1975) Zbl0325.46020
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  13. Kato T., Perturbation theory for linear operators, Springer-Verlag, 1980. Zbl0836.47009
  14. Martinon A., Cantidades operacionales en teoría de Fredholm (Thesis), Univ. La Laguna, 1989. MR1067932
  15. Pietsch A., Operator ideals, North-Holland, 1980. Zbl1012.47001MR0582655
  16. Rosenthal H.P., On totally incomparable Banach spaces, J. Funct. Anal. 4 (1969), 167-175. (1969) Zbl0184.15004MR0248506
  17. Schechter M., Quantities related to strictly singular operators, Indiana Univ. Math. J. 21 (1972), 1061-1071. (1972) Zbl0274.47007MR0295103
  18. Sedaev A.A., The structure of certain linear operators (in Russian), Mat. Issled. 5 (1970), 166-175, MR 43#2540, Zbl. 247#47005. (1970) MR0276800
  19. Stephani I., Operator ideals generalizing the ideal of strictly singular operators, Math. Nachrichten 94 (1980), 29-41. (1980) Zbl0455.47033MR0582517
  20. Weis L., Über striktly singuläre und striktly cosinguläre Operatoren in Banachräumen (Dissertation), Univ. Bonn, 1974. 
  21. Zemánek J., Geometric characteristics of semi-Fredholm operators and their asymptotic behaviour, Studia Math. 80 (1984), 219-234. (1984) MR0783991

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