Projections from onto

Author:
Pierre David Saphar

Journal:
Proc. Amer. Math. Soc. **127** (1999), 1127-1131

MSC (1991):
Primary 46B20; Secondary 46B28

MathSciNet review:
1473679

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Abstract: Let and 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 of continuous linear operators onto the space of compact linear operators.

b) .

The answer is positive in certain cases, in particular if or has an unconditional basis. It seems that there are few results in the direction of a general solution. For example, suppose that and are reflexive and that or has the approximation property. Then, if , there is no projection of norm 1, from onto . In this paper, one obtains, in particular, the following result:

**Theorem.** *Let be a real Banach space which is reflexive (resp. with a separable dual), of infinite dimension, and such that has the approximation property. Let be a real scalar with . Then can be equivalently renormed such that, for any projection from onto , one has . One gives also various results with two spaces and .*

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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

DOI:
https://doi.org/10.1090/S0002-9939-99-04645-6

Keywords:
Space of continuous linear operators,
space of compact operators,
projection

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

Article copyright:
© Copyright 1999
American Mathematical Society