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A direct proof of a theorem of West on sequences of Riesz operators

Published online by Cambridge University Press:  18 May 2009

Anthony F. Ruston
Anthony F. Ruston
Affiliation:
University College of North Wales, (Coleg Prifysgol Gogledd Cymru), Bangor, Caernarvonshire, LL57 2Uw, Wales
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We recall (cf. [2] Definitions 3.1 and 3.2, p. 322) that a bounded linear operator T on a Banach space ℵ into itself is said to be asymptotically quasi-compact if K(Tn)⅟n → 0 as n → ∞. where K(U) = inf ∥U–C∥ for every bounded linear operator U on ℵ into itself, the infimum being taken over all compact linear operators C on ℵ into itself. For a complex Banach space, this is equivalent (cf. [2], pp. 319, 321 and 326) to T being a Riesz operator.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 15 , Issue 2 , September 1974 , pp. 93 - 94
Copyright
Copyright © Glasgow Mathematical Journal Trust 1974

References

REFERENCES

1.Gillespie, T. A. and West, T. T., A characterisation and two examples of Riesz operators, Glasgow Math. J. 9 (1968), 106–110.CrossRefGoogle Scholar
2.Ruston, A. F., Operators with a Fredholm theory, J. London Math. Soc. 29 (1954), 318–326.CrossRefGoogle Scholar
3.West, T. T., Riesz operators in Banach spaces, Proc. London Math. Soc. (3) 16 (1966), 131–140.CrossRefGoogle Scholar
4.West, T. T., The decomposition of Riesz operators, Proc. London Math. Soc. (3) 16 (1966), 737–752.CrossRefGoogle Scholar