Projections from L ( X , Y ) onto K ( X , Y )

Kamil John

Commentationes Mathematicae Universitatis Carolinae (2000)

  • Volume: 41, Issue: 4, page 765-771
  • ISSN: 0010-2628

Abstract

top
Generalization of certain results in [Sap] and simplification of the proofs are given. We observe e.g.: Let X and Y be Banach spaces such that X is weakly compactly generated Asplund space and X * has the approximation property (respectively Y is weakly compactly generated Asplund space and Y * has the approximation property). Suppose that L ( X , Y ) ≠ K ( X , Y ) and let 1 < λ < 2 . Then X (respectively Y ) can be equivalently renormed so that any projection P of L ( X , Y ) onto K ( X , Y ) has the sup-norm greater or equal to λ .

How to cite

top

John, Kamil. "Projections from $L(X,Y)$ onto $K(X,Y)$." Commentationes Mathematicae Universitatis Carolinae 41.4 (2000): 765-771. <http://eudml.org/doc/248591>.

@article{John2000,
abstract = {Generalization of certain results in [Sap] and simplification of the proofs are given. We observe e.g.: Let $X$ and $Y$ be Banach spaces such that $X$ is weakly compactly generated Asplund space and $X^*$ has the approximation property (respectively $Y$ is weakly compactly generated Asplund space and $Y^*$ has the approximation property). Suppose that $L(X,Y)\ne K(X,Y)$ and let $1<\lambda <2$. Then $X$ (respectively $Y$) can be equivalently renormed so that any projection $P$ of $L(X,Y)$ onto $K(X,Y)$ has the sup-norm greater or equal to $\lambda $.},
author = {John, Kamil},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {compact operator; approximation property; reflexive Banach space; projection; separability; compact operator; approximation property; reflexive Banach space; projection},
language = {eng},
number = {4},
pages = {765-771},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Projections from $L(X,Y)$ onto $K(X,Y)$},
url = {http://eudml.org/doc/248591},
volume = {41},
year = {2000},
}

TY - JOUR
AU - John, Kamil
TI - Projections from $L(X,Y)$ onto $K(X,Y)$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2000
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 41
IS - 4
SP - 765
EP - 771
AB - Generalization of certain results in [Sap] and simplification of the proofs are given. We observe e.g.: Let $X$ and $Y$ be Banach spaces such that $X$ is weakly compactly generated Asplund space and $X^*$ has the approximation property (respectively $Y$ is weakly compactly generated Asplund space and $Y^*$ has the approximation property). Suppose that $L(X,Y)\ne K(X,Y)$ and let $1<\lambda <2$. Then $X$ (respectively $Y$) can be equivalently renormed so that any projection $P$ of $L(X,Y)$ onto $K(X,Y)$ has the sup-norm greater or equal to $\lambda $.
LA - eng
KW - compact operator; approximation property; reflexive Banach space; projection; separability; compact operator; approximation property; reflexive Banach space; projection
UR - http://eudml.org/doc/248591
ER -

References

top
  1. Amir D., Lindenstrauss J., The structure of weakly compact sets in Banach spaces, Ann. of Math. 88 (1968), 35-59. (1968) Zbl0164.14903MR0228983
  2. Arterburn D., Whitney R., Projections in the space of bounded linear operators, Pacific J. Math. 15 (1965), 739-746. (1965) MR0187052
  3. Casazza P.G., Kalton N.J., Notes on approximation properties on separable Banach spaces, in: Geometry of Banach spaces, London Math. Soc. Lecture Note Ser. 158, Cambridge Univ. Press, 1991, pp.49-63. MR1110185
  4. Diestel J., Morrison T.J., The Radon-Nikodym property for the space of operators, Math. Nachr. 92 (1979), 7-12. (1979) Zbl0444.46021MR0563569
  5. Emmanuele G., On the containment of c 0 by spaces of compact operators, Bull. Sci. Mat. 115 (1991), 177-184. (1991) MR1101022
  6. Emmanuele G., A remark on the containment of c 0 in spaces of compact operators, Math. Proc. Cambridge Phil. Soc. 111 (1992), 331-335. (1992) MR1142753
  7. Emmanuele G., John K., Uncomplementability of spaces of compact operators in lager spaces of operators, Czechoslovak J. Math. 47 (122) (1997), 19-32. (1997) MR1435603
  8. Feder M., On subspaces of spaces with an unconditional basis and spaces of operators, Illinois J. Math. 24 (1980), 196-205. (1980) Zbl0411.46009MR0575060
  9. Feder M., On the non-existence of a projection onto the spaces of compact operators, Canad. Math. Bull. 25 (1982), 78-81. (1982) MR0657655
  10. Godefroy G., Saphar P., Duality in spaces of operators and smooth norms on Banach spaces, Illinois J. Math. 32 (4) (1988), 672-695. (1988) Zbl0631.46015MR0955384
  11. Godun B.V., Unconditional bases and basic sequences, Izv. Vyssh. Uchebn. Zaved. Mat 24 (1980), 69-72. (1980) MR0603941
  12. John K., On the space K ( P , P * ) of compact operators on Pisier space P, Note di Matematica 72 (1992), 69-75. (1992) MR1258564
  13. John K., On the uncomplemented subspace 𝒦 ( X , Y ) , Czechoslovak Math. J. 42 (1992), 167-173. (1992) MR1152178
  14. Johnson J., Remarks on Banach spaces of compact operators, J. Funct. Analysis 32 (1979), 304-311. (1979) Zbl0412.47024MR0538857
  15. Kuo T.H., Projections in the space of bounded linear operators, Pacific. J. Math. 52 (1974), 475-480. (1974) MR0352939
  16. Kalton N.J., Spaces of compact operators, Math. Ann. 208 (1974), 267-278. (1974) Zbl0266.47038MR0341154
  17. Lindenstrauss J., On nonseparable reflexive Banach spaces, Bull. Amer. Math. Soc. 72 (1966), 967-970. (1966) Zbl0156.36403MR0205040
  18. Ruess W., Duality and geometry of spaces of compact operators, in Functional Analysis: Surveys and Recent Results III, Math. Studies 90, North Holland, 1984. Zbl0573.46007MR0761373
  19. Saphar P.D., Projections from L ( E , F ) onto K ( E , F ) , Proc. Amer. Math. Soc. 127 (4) (1999), 1127-1131. (1999) Zbl0912.46011MR1473679
  20. Singer I., Bases in Banach spaces. Vol. II, Berlin-Heidelberg-New York, Springer, 1981. MR0610799
  21. Thorp E., Projections onto the space of compact operators, Pacific J. Math. 10 (1960), 693-696. (1960) MR0114128
  22. Tong A.E., On the existence of non-compact bounded linear operators between certain Banach spaces, Israel J. Math. 10 (1971), 451-456. (1971) MR0296663
  23. Tong A.E., Wilken D.R., The uncomplemented subspace K ( E , F ) , Studia Math. 37 (1971), 227-236. (1971) Zbl0212.46302MR0300058
  24. Zippin M., Banach spaces with separable duals, Trans. Amer. Math. Soc. 310 (1988), 371-379. (1988) Zbl0706.46015MR0965758

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.