Hostname: page-component-857557d7f7-zv5th Total loading time: 0 Render date: 2025-11-20T06:08:59.484Z Has data issue: false hasContentIssue false
  • English
  • Français

Operators with large trace, and a characterization of

Published online by Cambridge University Press:  24 October 2008

D. J. H. Garling
D. J. H. Garling
Affiliation:
St John's College, Cambridge
Get access Rights & Permissions [Opens in a new window]

Extract

Let E be a real or complex n-dimensional normed space. Deschaseaux (1) has shown that if the absolutely summing constant π1(E) = n then E is isometrically isomorphic to . (For the definition of π1(E) and related quantities, see (2).) It is well-known ((3), (4) and (6)) that if E is a P1-space then E is isometrically isomorphic to . We shall show that Deschaseaux' result follows easily from this, and from the following elementary result:Proposition. If S is a linear operator on a real or complex n-dimensional normed space E, then |trace S| ≤ ‖S‖, with equality if and only if S is a scalar multiple of the identity I.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 76 , Issue 2 , September 1974 , pp. 413 - 414
Copyright
Copyright © Cambridge Philosophical Society 1974

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

(1)Deschaseaux, J. P.Une caractérisation de certains espaces vectoriels normés do dimension finie par leur constante do MacPhail. C.R. Acad. Sci. Paris Sér. A–B 276 (1973), A1349–A1351.Google Scholar
(2)Garling, D. J. H. and Gordon, Y.Relations between some constants associated with finite dimensional Banach spaces. Israel J. Math. 9 (1971), 346–361.CrossRefGoogle Scholar
(3)Goodner, D. B.Projections in normed linear spaces. Trans. Amer. Math. Soc. 69 (1950), 89–108.CrossRefGoogle Scholar
(4)Kelley, J. L.Banach spaces with the extension property. Trans. Amer. Math. Soc. 72 (1952), 323–326.CrossRefGoogle Scholar
(5)Lindenstrauss, J.Extension of compact operators. Mem. Amer. Math. Soc. 48 (1964).Google Scholar
(6)Nachbin, L.A theorem of the Hahn–Banach type for linear transformations. Trans. Amer. Math. Soc. 68 (1950), 28–46.CrossRefGoogle Scholar