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Core-preserving transformations of a vector space

Published online by Cambridge University Press:  24 October 2008

F. F. Bonsall
F. F. Bonsall
Affiliation:
King's CollegeDurham UniversityNewcastle upon Tyne
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Extract

In the classical theory (3) due to Knopp, Agnew and others, the core K(x) of a sequence x = {ξn} of complex numbers is defined by where En(x) is the smallest closed convex set containing all ξk with k ≥ n. A matrix transformation T is said to be a core-preserving transformation if

holds for all sequences x. T is core-preserving for bounded sequences if (1·1) holds for all bounded sequences x. It is readily proved that K(x) is the set of complex numbers ζ such that

for all complex numbers α (). Now is a sub-additive, positive-homogeneous real-valued functional defined on the vector space of bounded complex sequences. This suggests the construction of an abstract theory on the following lines.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 49 , Issue 1 , January 1953 , pp. 15 - 25
Copyright
Copyright © Cambridge Philosophical Society 1953

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References

REFERENCES

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