The generalized Fredholm operators

Yang, Kung Wei

doi:10.1090/S0002-9947-1976-0423114-X

The generalized Fredholm operators
HTML articles powered by AMS MathViewer

by Kung Wei Yang PDF

Trans. Amer. Math. Soc. 216 (1976), 313-326 Request permission

Abstract:

Let X, Y be Banach spaces over either the real field or the complex field. A continuous linear operator will be called a generalized Fredholm operator if $T(X)$ is closed in Y, and Ker T and Coker T are reflexive Banach spaces. A theory similar to the classical Fredholm theory exists for the generalized Fredholm operators; and the similarity brings out the correspondence: Reflexive Banach spaces $\leftrightarrow$ finite-dimensional spaces, weakly compact operators $\leftrightarrow$ compact operators, generalized Fredholm operators $\leftrightarrow$ Fredholm operators, Tauberian operators with closed range $\leftrightarrow$ semi-Fredholm operators.

References

Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
Paul Civin and Bertram Yood, Quasi-reflexive spaces, Proc. Amer. Math. Soc. 8 (1957), 906–911. MR 90020, DOI 10.1090/S0002-9939-1957-0090020-6
Mahlon M. Day, Normed linear spaces, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21, Springer-Verlag, New York-Heidelberg, 1973. MR 0344849

Fondements de l’analyse moderne

22

Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
Nigel Kalton and Albert Wilansky, Tauberian operators on Banach spaces, Proc. Amer. Math. Soc. 57 (1976), no. 2, 251–255. MR 473896, DOI 10.1090/S0002-9939-1976-0473896-1
Saunders MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York-Berlin, 1971. MR 0354798
Richard S. Palais, Seminar on the Atiyah-Singer index theorem, Annals of Mathematics Studies, No. 57, Princeton University Press, Princeton, N.J., 1965. With contributions by M. F. Atiyah, A. Borel, E. E. Floyd, R. T. Seeley, W. Shih and R. Solovay. MR 0198494, DOI 10.1515/9781400882045
N. Popescu, Abelian categories with applications to rings and modules, London Mathematical Society Monographs, No. 3, Academic Press, London-New York, 1973. MR 0340375
Robert Schatten, A Theory of Cross-Spaces, Annals of Mathematics Studies, No. 26, Princeton University Press, Princeton, N. J., 1950. MR 0036935
Martin Schechter, Principles of functional analysis, Academic Press, New York-London, 1971. MR 0445263
Kung-wei Yang, A note on reflexive Banach spaces, Proc. Amer. Math. Soc. 18 (1967), 859–861. MR 215061, DOI 10.1090/S0002-9939-1967-0215061-3
Kung-wei Yang, Completion of normed linear spaces, Proc. Amer. Math. Soc. 19 (1968), 801–806. MR 226372, DOI 10.1090/S0002-9939-1968-0226372-0
Kung Wei Yang, Index of Fredholm operators, Proc. Amer. Math. Soc. 41 (1973), 329–330. MR 318946, DOI 10.1090/S0002-9939-1973-0318946-5
Ronald G. Douglas, Banach algebra techniques in operator theory, Pure and Applied Mathematics, Vol. 49, Academic Press, New York-London, 1972. MR 0361893