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On Spaces of Compact Operators in Non-Archimedean Banach Spaces

Published online by Cambridge University Press:  20 November 2018

Takemitsu Kiyosawa
Takemitsu Kiyosawa*
Affiliation:
Faculty of Education Shizuoka University Ohya, Shizuoka, 422 Japan
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Abstract

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Let K be a non-trivial complete non-Archimedean valued field and let E be an infinite-dimensional Banach space over K. Some of the main results are:

(1) K is spherically complete if and only if every weakly convergent sequence in l∞ is norm-convergent.

(2) If the valuation of K is dense, then C0 is complemented in E if and only if C(E,c0) is n o t complemented in L(E,c0), where L(E,c0) is the space of all continuous linear operators from E to c0 and C(E,c0) is the subspace of L(E, c0) consisting of all compact linear operators.

Keywords

46P0530G0512J25non-Archimedean Banach spacesspherically complete non-Archimedean valued fieldscompact operators
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 32 , Issue 4 , 01 December 1989 , pp. 450 - 458
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Amice, Y., Interpolation p-adique, Bull. Soc. Math. France, 92 (1964), 117–180.Google Scholar
2. de, N. Grande-de Kimpe, Structure theorems for locally K-convex spaces, Indag. Math., 39 (1977), 11–22.Google Scholar
3. Gruson, L., Théorie de Fredholm p-adique, Bull. Soc. Math. France, 94 (1966), 67–95.Google Scholar
4. Josef, B. on, Weak sequential convergence in the dual of a Banach space does not imply norm convergence, Ark. Math., 13 (1975), 79–89.Google Scholar
5. Kiyosawa, T., Banach's closed range theorem and Fredholm alternative theorem in non-Archimedean Banach spaces, Illinois J. Math., 28 (1984), 353–369.Google Scholar
6. Kuo, T. H., Projections in the spaces of bounded linear operators, Pacific J. Math., 52 (1974), 475–480.Google Scholar
7. Monna, A. F., Analyse non-Archimedienne, Springer-Verlag, Berlin-Heidelberg, 1970.Google Scholar
8. Narici, L., Beckenstein, E. and Bachman, G., Fucntional analysis and valuation theory, Marcel Dekker, Inc., New York, 1971.Google Scholar
9. Nissenzweig, A., w* sequential convergence, Israel J. Math. 22 (1975), 266–272.Google Scholar
10. van Rooij, A., Non-Archimedean functional analysis, Marcel Dekker, Inc., New York, 1978.Google Scholar
11. Schikhof, W. H., Ultrametric calculus, Cambridge University Press, 1984.Google Scholar
12. Serre, J. P., Endomorphisms complètement continus des espaces de Banach p-adiques, Inst. Hautes Études Sci. Publ. Math., 12 (1962), 69–85.Google Scholar
13. Thorp, E. O., Projections onto the subspace of compact operators, Pacific J. Math. 10 (1960), 693–696.Google Scholar
14. Tong, A. E. and D. R. Wilken, The uncomplemented subspace K(E,F), Studia Math., 37 (1971), 227–236.Google Scholar