Spectrum (linear operator): Difference between revisions
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A bounded, linear operator ''O'' that maps a [[Banach space]] into itself has a '''spectrum''' of values {''λ''} provided there are non-zero vectors ''x<sub>λ</sub>'' in the space such that {{nowrap|''O x<sub>λ</sub>''<nowiki> =</nowiki> ''λ x<sub>λ</sub>''.}} The {''λ''} are called ''characteristic values'' of ''O'' and {''x<sub>λ</sub>''} the ''eigenvectors'' of ''O''. The spectrum may consist of discrete values, a continuum of values, or a combination of both.<ref name=Fomin/> | |||
==References== | |||
{{reflist |refs= | |||
<ref name=Fomin> | |||
{{cite book |title=Elements of the Theory of Functions and Functional Analysis, Volume 1 |author=A. N. Kolmogorov, Sergeĭ Vasilʹevich Fomin, S. V. Fomin |url=http://books.google.com/books?id=OyWeDwfQmeQC&pg=PA110 |pages=pp. 110 ''ff'' |chapter=§30 Spectrum of an operator. Resolvents |isbn=0486406830 |year=1999 |publisher=Courier Dover Publications |edition=Reprint of Graylock Press 1957 ed}} | |||
</ref> | |||
}} | |||
Latest revision as of 11:06, 13 June 2012
A bounded, linear operator O that maps a Banach space into itself has a spectrum of values {λ} provided there are non-zero vectors xλ in the space such that O xλ = λ xλ. The {λ} are called characteristic values of O and {xλ} the eigenvectors of O. The spectrum may consist of discrete values, a continuum of values, or a combination of both.[1]
References
- ↑ A. N. Kolmogorov, Sergeĭ Vasilʹevich Fomin, S. V. Fomin (1999). “§30 Spectrum of an operator. Resolvents”, Elements of the Theory of Functions and Functional Analysis, Volume 1, Reprint of Graylock Press 1957 ed. Courier Dover Publications, pp. 110 ff. ISBN 0486406830.