Projections from onto

Author:
Pierre David Saphar

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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 .*

**1.**P. G. Casazza and N. J. Kalton,*Notes on approximation properties on separable Banach spaces*, in: Geometry of Banach Spaces, London Math. Soc. Lecture Note Ser. 158, Cambridge Univ. Press (1991), 49-63. MR**92d:46022****2.**J. Dixmier,*Sur un théorème de Banach*, Duke Math. J.**15**(1948), 1057-1071. MR**10:306g****3.**G. Godefroy and P. D. Saphar,*Duality in spaces of operators and smooth norms on Banach spaces*, Illinois J. Math.**32**(1988), 673-695. MR**89j:47026****4.**G. Godefroy, N. J. Kalton and P. D. Saphar,*Unconditional ideals in Banach spaces*, Studia Math.**104**(1993), 13-59. MR**94k:46024****5.**K. John,*On the uncomplemented subspace*, Czechoslovak Math. J.**42**(117) (1992), 167-173. MR**93b:47085****6.**J. Lindenstrauss,*On nonseparable reflexive Banach spaces*, Bull. Amer. Math. Soc.**72**(1966), 967-970. MR**34:4875****7.**J. Lindenstrauss and L. Tzafriri,*Classical Banach spaces*, vol. I, Sequence spaces, Springer-Verlag, Berlin, 1977. MR**58:17766****8.**E. Odell and H. P. Rosenthal,*A double dual characterization of separable Banach spaces containing*, Israel J. Math.**20**(1975), 375-384. MR**51:13654****9.**H. H. Schaefer,*Banach lattices and positive operators*, Springer-Verlag, New York, 1974. MR**54:11023****10.**M. Zippin,*Banach spaces with separable duals*, Trans. Amer. Math. Soc.**310**(1988), 371-379. MR**90b:46028****11.**B. V. Godun,*Unconditional bases and spanning basic sequences*, Izv. Vyssh. Uchebn. Zaved. Mat.**24**(1980), 69-72. MR**82h:46017**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
46B20,
46B28

Retrieve articles in all journals with MSC (1991): 46B20, 46B28

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