Projections from L(E,F) onto K(E,F)
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- by Pierre David Saphar PDF
- Proc. Amer. Math. Soc. 127 (1999), 1127-1131 Request permission
Abstract:
Let $E$ and $F$ be two infinite dimensional real Banach spaces. The following question is classical and long-standing. Are the following properties equivalent? a) There exists a projection from the space $L(E,F)$ of continuous linear operators onto the space $K(E,F)$ of compact linear operators. b) $L(E,F)=K(E,F)$. The answer is positive in certain cases, in particular if $E$ or $F$ has an unconditional basis. It seems that there are few results in the direction of a general solution. For example, suppose that $E$ and $F$ are reflexive and that $E$ or $F$ has the approximation property. Then, if $L(E,F)\ne K(E,F)$, there is no projection of norm 1, from $L(E,F)$ onto $K(E,F)$. In this paper, one obtains, in particular, the following result: Theorem. Let $F$ be a real Banach space which is reflexive (resp. with a separable dual), of infinite dimension, and such that $F^*$ has the approximation property. Let $\lambda$ be a real scalar with $1<\lambda <2$. Then $F$ can be equivalently renormed such that, for any projection $P$ from $L(F)$ onto $K(F)$, one has $\|P\|\ge \lambda$. One gives also various results with two spaces $E$ and $F$.References
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Additional Information
- Pierre David Saphar
- Affiliation: Department of Mathematics, Technion–Israel Institute of Technology, Haifa, Israel
- Email: saphar@techunix.technion.ac.il
- Received by editor(s): October 30, 1996
- Received by editor(s) in revised form: July 28, 1997
- Additional Notes: This research was supported by the fund for the promotion of Research at the Technion
- Communicated by: Dale E. Alspach
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 1127-1131
- MSC (1991): Primary 46B20; Secondary 46B28
- DOI: https://doi.org/10.1090/S0002-9939-99-04645-6
- MathSciNet review: 1473679