Generalized Fredholm operators and the boundary of the maximal group of invertible operators
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- by G. W. Treese and E. P. Kelly PDF
- Proc. Amer. Math. Soc. 67 (1977), 123-128 Request permission
Abstract:
Let V denote an infinite dimensional Banach space over the complex field and let $G[V]$ denote the subset of bounded operators on V with the property that the null space has a closed complement and the range is closed, where the null space and range are proper subspaces of V. Necessary and sufficient conditions for $T \in G[V]$ to be in the boundary, $\mathcal {B}$, of the maximal group, $\mathcal {M}$, of invertible operators are determined. As a result, $\mathcal {B} \cap G[V]$ is the set of products of operators in $\mathcal {M}$ and operators in $\mathcal {P}$, where $\mathcal {P}$ is the set of projections other than the identity operator and null operator.References
- F. V. Atkinson, On relatively regular operators, Acta Sci. Math. (Szeged) 15 (1953), 38–56. MR 56835
- George Bachman and Lawrence Narici, Functional analysis, Academic Press, New York-London, 1966. MR 0217549
- Frederick J. Beutler, The operator theory of the pseudo-inverse. I. Bounded operators, J. Math. Anal. Appl. 10 (1965), 451–470. MR 179618, DOI 10.1016/0022-247X(65)90108-3
- S. R. Caradus, Perturbation theory for generalized Fredholm operators, Pacific J. Math. 52 (1974), 11–15. MR 353034
- Jacob Feldman and Richard V. Kadison, The closure of the regular operators in a ring of operators, Proc. Amer. Math. Soc. 5 (1954), 909–916. MR 68749, DOI 10.1090/S0002-9939-1954-0068749-2
- D. A. Hogan and C. E. Langenhop, Invertibility in a Banach algebra, Indiana Univ. Math. J. 24 (1974/75), 965–977. MR 370192, DOI 10.1512/iumj.1975.24.24081
- E. P. Kelly Jr. and D. A. Hogan, Bounded, conservative, linear operators and the maximal group. II, Proc. Amer. Math. Soc. 38 (1973), 298–302. MR 313832, DOI 10.1090/S0002-9939-1973-0313832-9 M. Z. Nashed and G. F. Votruba, A united approach to generalized inverses of linear operators, Bull. Amer. Math. Soc. 80 (1974), 825-830, 831-835.
- B. E. Rhoades, Triangular summability methods and the boundary of the maximal group, Math. Z. 105 (1968), 284–290. MR 228882, DOI 10.1007/BF01125969
- Martin Schechter, Principles of functional analysis, Academic Press, New York-London, 1971. MR 0445263
- Angus E. Taylor, Introduction to functional analysis, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 0098966
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 67 (1977), 123-128
- MSC: Primary 47B30; Secondary 47A99
- DOI: https://doi.org/10.1090/S0002-9939-1977-0454712-1
- MathSciNet review: 0454712