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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
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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.

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

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References

REFERENCES

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