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Fredholm operator families -I

Abstract

First, we give a characterization of semi-Fredholm operators (i.e. those which are left or right invertible modulo compact operators) on Hausdorff tvs which generalizes the usual one in the context of Banach spaces. Then we consider a class of semi-Fredholm operator families on tvs which include both the "classical" semi-Fredholm operator valued functions on Banach spaces (continuous in the norm sense), and families of the form T + Kn, where Kn is a collectively compact sequence which converges strongly to O. For these families we prove a general stability theorem.

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References

  1. 1

    Adasch, N., Keim, B., Ernst, D.: Topological Vector Spaces. The Theory without Convexity Conditions. Springer-Verlag. LNM 639. Berlin, Heidelberg, New York (1978).

  2. 2.

    Anselone, P.M.: Collectively Compact Operator Approximation and Applications to Integral Equations. Prentice-Hall, Englewood Cliffs, New Jersey (1971).

  3. 3.

    Berger, R.: Fredholmoperatoren in separierten topologischen Vektorräumen und lokalbeschränkten Räumen. Diplomarbeit. Johannes Gutenberg-Universität. Mainz (1969).

  4. 4.

    Cuellar, J.: Fredholmoperatoren auf lokalbeschränkten Räumen mit Anwendungen auf elliptische Gleichungen. Dissertation. Johannes Gutenberg-Universität. Mainz (1982).

  5. 5.

    Deprit, A.: Quelques classes d'endomorphismes d'espaces localment convex séparés. Bull. el. sci. Acad. Roy. Belg., 43, 252–272 (1975).

  6. 6.

    De Pree, J.D., Higgins, J.A.: Collectively Compact Sets of Linear Operators. Math. Z. 115, 366–370 (1970).

  7. 7.

    Dolganova, S.: Semi-Fredholm Operator Families on Topological Vector Spaces. (Russian). Uspekhi Mat. Nauk, 27:4, 211–212 (1972).

  8. 8.

    Goldberg, S.: Perturbations of Semi-Fredholm Operators by Operators Converging to Zero Compactly. Proc. Amer. Math. Soc. 45, 93–98 (1974).

  9. 9.

    Gramsch, B.: Ein Schema zur Theorie Fredholmscher Endomorphismen und eine Anwendung auf die Idealkette der Hilberträume. Math. Annalen 171, 263–272 (1967).

  10. 10.

    Gramsch, B., Kaballo, W.: Spectral Theory for Fredholm Operators. In: Functional Analysis: Surveys and Recent Results II, 319–342. Ed.: K.D. Bierstedt, B. Fuchssteiner. North-Holland. Amsterdam (1980).

  11. 11.

    Hörmander, L.: On the Index of Pseudodiffential Operators. In: Elliptische Differentialgleichungen. Band II. 127–146. Akademie-Verlag. Berlin (1971).

  12. 12.

    Kaballo, W.: Holomorphe Semifredholm-Operatorfunktionen in lokalkonvexen Räumen. Diplomarbeit. Kaiserslautern (1973).

  13. 13.

    Kaballo, W.: Holomorphe Semifredholmfunktionen mit Anwendungen auf Differential- und Pseudodifferentialoperatoren. Dissertation. Kaiserslautern (1974).

  14. 14.

    Pietsch, A.: Zur Theorie der σ-Transformationen in lokalkonvexen Vektorräumen. Math. Nachr. 21, 347–369 (1960).

  15. 15.

    Segal, G.: Fredholm Complexes. Quart. J. Math. Oxford (2), 21, 385–402 (1970).

  16. 16.

    Sobolev, S.L.: Some remarks about the numerical solution of integral equations. Izvestia Acad. Sci. USSR, vol 20, n° 4, 413–436 (1956).

  17. 17.

    Vainikko, G.M.: Regular Convergence of Operators and Approximate Solution of Equations. Jour. of Sov. Math. 15 (6), 675–705 (1981). Translated from Itogi Nauki i Tekhniki, Seriya Mat. Analiz, 16, 5–53 (1979).

  18. 18.

    Williamson. J.H.: Compact Linear Operators in Linear Topological Spaces. Jour. Lond. Math. Soc. 29, 149–156 (1954).

  19. 19.

    Zaidenberg, M.G., Krein S.G., Kuchment, P.A., Pankov, A.A.: Banach Bundles and Linear Operators. Russian Math. Surveys 30:5, 115–175 (1975). Translated from Uspekhi mat. Nauk 30:5, 101–157 (1975).

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Cuellar, J., Dynin, A. & Dynin, S. Fredholm operator families -I . Integr equ oper theory 6, 853–862 (1983). https://doi.org/10.1007/BF01691927

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Keywords

  • Banach Space
  • Operator Family
  • Compact Operator
  • General Stability
  • Fredholm Operator