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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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 $F$.
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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