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Article: Numerical ranges of an operator on an indefinite inner product space

TitleNumerical ranges of an operator on an indefinite inner product space
Authors
KeywordsConvexity
Field Of Values
Generalized Numerical Range
Indefinite Inner Product Space
Krein Space
Linear Preserver
Numerical Range
Issue Date1996
PublisherInternational Linear Algebra Society. The Journal's web site is located at http://www.math.technion.ac.il/iic/ela
Citation
Electronic Journal Of Linear Algebra, 1996, v. 1, p. 1-17 How to Cite?
AbstractFor n x n complex matrices A and an n x n Hermitian matrix S, we consider the S-numerical range of A and the positive S-numerical range of A defined by WS(A) = {〈Av, v〉S/〈v, v〉S : v ∈ ℂn, 〈v, v〉S ≠ 0} and W S + (A) = {〈Av, v〉S : v ∈ ℂn, 〈v, v〉S = 1}, respectively, where 〈u, v〉S = v*Su. These sets generalize the classical numerical range, and they are closely related to the joint numerical range of three Hermitian forms and the cone generated by it. Using some theory of the joint numerical range we can give a detailed description of WS(A) and WS + (A) for arbitrary Hermitian matrices S. In particular, it is shown that WS + (A) is always convex and WS(A) is always p-convex for all S. Similar results are obtained for the sets VS(A) = {〈Av, v〉/〈Sv, v〉: v ∈ ℂn, 〈Sv, v〉 ≠ 0}, VS + (A) = {〈Av, v〉: v ∈ ℂn, 〈Sv, v〉 = 1}, where 〈u, v〉 = v* u. Furthermore, we characterize those linear operators preserving WS(A), WS + (A), V S(A), or VS + (A). Possible generalizations of our results, including their extensions to bounded linear operators on an infinite dimensional Hilbert or Krein space, are discussed.
Persistent Identifierhttp://hdl.handle.net/10722/156044
ISSN
2010 Impact Factor: 0.808
2015 SCImago Journal Rankings: 0.566
References

 

DC FieldValueLanguage
dc.contributor.authorLi, CKen_US
dc.contributor.authorTsing, NKen_US
dc.contributor.authorUhlig, Fen_US
dc.date.accessioned2012-08-08T08:40:10Z-
dc.date.available2012-08-08T08:40:10Z-
dc.date.issued1996en_US
dc.identifier.citationElectronic Journal Of Linear Algebra, 1996, v. 1, p. 1-17en_US
dc.identifier.issn1081-3810en_US
dc.identifier.urihttp://hdl.handle.net/10722/156044-
dc.description.abstractFor n x n complex matrices A and an n x n Hermitian matrix S, we consider the S-numerical range of A and the positive S-numerical range of A defined by WS(A) = {〈Av, v〉S/〈v, v〉S : v ∈ ℂn, 〈v, v〉S ≠ 0} and W S + (A) = {〈Av, v〉S : v ∈ ℂn, 〈v, v〉S = 1}, respectively, where 〈u, v〉S = v*Su. These sets generalize the classical numerical range, and they are closely related to the joint numerical range of three Hermitian forms and the cone generated by it. Using some theory of the joint numerical range we can give a detailed description of WS(A) and WS + (A) for arbitrary Hermitian matrices S. In particular, it is shown that WS + (A) is always convex and WS(A) is always p-convex for all S. Similar results are obtained for the sets VS(A) = {〈Av, v〉/〈Sv, v〉: v ∈ ℂn, 〈Sv, v〉 ≠ 0}, VS + (A) = {〈Av, v〉: v ∈ ℂn, 〈Sv, v〉 = 1}, where 〈u, v〉 = v* u. Furthermore, we characterize those linear operators preserving WS(A), WS + (A), V S(A), or VS + (A). Possible generalizations of our results, including their extensions to bounded linear operators on an infinite dimensional Hilbert or Krein space, are discussed.en_US
dc.languageengen_US
dc.publisherInternational Linear Algebra Society. The Journal's web site is located at http://www.math.technion.ac.il/iic/elaen_US
dc.relation.ispartofElectronic Journal of Linear Algebraen_US
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.subjectConvexityen_US
dc.subjectField Of Valuesen_US
dc.subjectGeneralized Numerical Rangeen_US
dc.subjectIndefinite Inner Product Spaceen_US
dc.subjectKrein Spaceen_US
dc.subjectLinear Preserveren_US
dc.subjectNumerical Rangeen_US
dc.titleNumerical ranges of an operator on an indefinite inner product spaceen_US
dc.typeArticleen_US
dc.identifier.emailTsing, NK:nktsing@hku.hken_US
dc.identifier.authorityTsing, NK=rp00794en_US
dc.description.naturepublished_or_final_versionen_US
dc.identifier.scopuseid_2-s2.0-0003084951en_US
dc.identifier.hkuros20725-
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-0003084951&selection=ref&src=s&origin=recordpageen_US
dc.identifier.volume1en_US
dc.identifier.spage1en_US
dc.identifier.epage17en_US
dc.publisher.placeUnited Statesen_US
dc.identifier.scopusauthoridLi, CK=8048590800en_US
dc.identifier.scopusauthoridTsing, NK=6602663351en_US
dc.identifier.scopusauthoridUhlig, F=7003921862en_US

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