Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-29T12:43:38.055Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Non-Existence of a Projection onto the Space of Compact Operators

Published online by Cambridge University Press:  20 November 2018

Moshe Feder
Moshe Feder*
Affiliation:
Department of Mathematics University of Toronto, Toronto, Canada M5S 1A1
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let X and Y be Banach spaces, L(X, Y) the space of bounded linear operators from X to Y and C(X, Y) its subspace of the compact operators. A sequence {Ti} in C(X, Y) is said to be an unconditional compact expansion of T ∈ L (X, Y) if ∑ Tix converges unconditionally to Tx for every x ∈ X. We prove: (1) If there exists a non-compact T ∈ L(X, Y) admitting an unconditional compact expansion then C(X, Y) is not complemented in L(X, Y), and (2) Let X and Y be classical Banach spaces (i.e. spaces whose duals are some LP(μ) spaces) then either L(X, Y) = C(X, Y) or C(X, Y) is not complemented in L(X, Y).

Keywords

46A3246B2046B2547D15Banach spacescompact operatorsprojectionsunconditional compact expansion
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 25 , Issue 1 , 01 March 1982 , pp. 78 - 81
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Arterburn, D. and Whitley, R. J., Projections in the space of bounded linear operators. Pacific J. Math. 15 (1965), 739-746.Google Scholar
2. Bessaga, C. and PeƗczyński, A., On bases and unconditional convergence of series in Banach spaces. Studia Math. 17 (1958), 151-174.Google Scholar
3. Day, M. M., Normed Linear Spaces (3rd Edition), Springer, New York, 1973.Google Scholar
4. Feder, M., Subspaces of spaces with an unconditional basis and spaces of operators. Illinois J. Math. 24 (1980), 196-205.Google Scholar
5. Johnson, J., Remarks on Banach spaces of operators. J. Functional Anal. 32 (1979), 304-311.Google Scholar
6. Kalton, N. J., Spaces of compact operators. Math. Ann. 208 (1974), 267-278.Google Scholar
7. Kuo, T., Projections in the spaces of bounded linear operators. Pacific J. Math. 52 (1974), 475-480.Google Scholar
8. Lacey, H. E., The Isometric Theory of Classical Banach Spaces. Springer, New York, 1974.Google Scholar
9. Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces I. Springer, New York, 1977.Google Scholar
10. Rosenthal, H. P., On quasi-complemented subspaces of Banach spaces, with an appendix on compactness of operators from Lp(μ) to Lr(v). J. Functional Anal. 4 (1969), 176-214.Google Scholar
11. Singer, I., Bases in Banach Spaces I. Springer, New York, 1970.Google Scholar
12. Thorp, E., Projections onto the subspace of compact operators. Pacific J. Math. 10 (1960), 693-696.Google Scholar
13. Tong, A. E. and Wilken, D. R., The uncomplemented subspace K(E,F). Studia Math. 37(1971), 227-236.Google Scholar
14. Zippin, M., On some subspaces of Banach spaces whose duals are L1 spaces. Proc. Amer. Math. Soc. 23 (1969), 378-385.Google Scholar