Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-15T10:22:51.390Z Has data issue: false hasContentIssue false

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
Affiliation:
University College of North Wales, (Coleg Prifysgol Gogledd Cymru), Bangor, Caernarvonshire, LL57 2Uw, Wales
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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)n → 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
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), 106110.CrossRefGoogle Scholar
2.Ruston, A. F., Operators with a Fredholm theory, J. London Math. Soc. 29 (1954), 318326.CrossRefGoogle Scholar
3.West, T. T., Riesz operators in Banach spaces, Proc. London Math. Soc. (3) 16 (1966), 131140.CrossRefGoogle Scholar
4.West, T. T., The decomposition of Riesz operators, Proc. London Math. Soc. (3) 16 (1966), 737752.CrossRefGoogle Scholar