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Article: G-invariant norms and G(c)-radii

TitleG-invariant norms and G(c)-radii
Authors
Issue Date1991
PublisherElsevier Inc. The Journal's web site is located at http://www.elsevier.com/locate/laa
Citation
Linear Algebra And Its Applications, 1991, v. 150 C, p. 179-194 How to Cite?
AbstractLet V be a finite dimensional inner product space over F(=R or C), and let G be a closed subgroup of the group of unitary operators on V. A norm or a seminorm ∥·∥ on V is said to be G-invariant if {norm of matrix}g(x){norm of matrix}=∥x∥ for all g ε{lunate} G, x ε{lunate} V. The concept of G-invariant norm specializes to many interesting particular cases such as the absolute norms on Fn, symmetric gauge functions on Rn, unitarily invariant norms on Fm×n, etc., which are of wide research interest. In this paper, we study the general properties of G-invariant norms. Our main strategy is to study G-invariant norms via the G(c)-radius rG(c)(·) on V defined by rG(c)(x) = max{|〈x, g(c)〉|:gε{lunate} G}, where c ε{lunate} V. It is shown that the G(c)-radii are very important G-invariant seminorms because every G-invariant norm or seminorm admits a representation in terms of them. As a result, one may focus attention on G(c)-radii in order to get results on G-invariant norms. We study the norm properties of G(c)-radii and obtain various results relating G-invariant norms and G(c)-radii. The linear operators on V that preserve G-invariant norms, G-invariant seminorms, or G(c)-radii are also investigated. Several open questions are mentioned. © 1991.
Persistent Identifierhttp://hdl.handle.net/10722/156108
ISSN
2015 Impact Factor: 0.965
2015 SCImago Journal Rankings: 0.837
ISI Accession Number ID

 

DC FieldValueLanguage
dc.contributor.authorLi, CKen_US
dc.contributor.authorTsing, NKen_US
dc.date.accessioned2012-08-08T08:40:26Z-
dc.date.available2012-08-08T08:40:26Z-
dc.date.issued1991en_US
dc.identifier.citationLinear Algebra And Its Applications, 1991, v. 150 C, p. 179-194en_US
dc.identifier.issn0024-3795en_US
dc.identifier.urihttp://hdl.handle.net/10722/156108-
dc.description.abstractLet V be a finite dimensional inner product space over F(=R or C), and let G be a closed subgroup of the group of unitary operators on V. A norm or a seminorm ∥·∥ on V is said to be G-invariant if {norm of matrix}g(x){norm of matrix}=∥x∥ for all g ε{lunate} G, x ε{lunate} V. The concept of G-invariant norm specializes to many interesting particular cases such as the absolute norms on Fn, symmetric gauge functions on Rn, unitarily invariant norms on Fm×n, etc., which are of wide research interest. In this paper, we study the general properties of G-invariant norms. Our main strategy is to study G-invariant norms via the G(c)-radius rG(c)(·) on V defined by rG(c)(x) = max{|〈x, g(c)〉|:gε{lunate} G}, where c ε{lunate} V. It is shown that the G(c)-radii are very important G-invariant seminorms because every G-invariant norm or seminorm admits a representation in terms of them. As a result, one may focus attention on G(c)-radii in order to get results on G-invariant norms. We study the norm properties of G(c)-radii and obtain various results relating G-invariant norms and G(c)-radii. The linear operators on V that preserve G-invariant norms, G-invariant seminorms, or G(c)-radii are also investigated. Several open questions are mentioned. © 1991.en_US
dc.languageengen_US
dc.publisherElsevier Inc. The Journal's web site is located at http://www.elsevier.com/locate/laaen_US
dc.relation.ispartofLinear Algebra and Its Applicationsen_US
dc.titleG-invariant norms and G(c)-radiien_US
dc.typeArticleen_US
dc.identifier.emailTsing, NK:nktsing@hku.hken_US
dc.identifier.authorityTsing, NK=rp00794en_US
dc.description.naturelink_to_subscribed_fulltexten_US
dc.identifier.doi10.1016/0024-3795(91)90168-V-
dc.identifier.scopuseid_2-s2.0-0040901914en_US
dc.identifier.volume150en_US
dc.identifier.issueCen_US
dc.identifier.spage179en_US
dc.identifier.epage194en_US
dc.identifier.isiWOS:A1991FE28000013-
dc.publisher.placeUnited Statesen_US
dc.identifier.scopusauthoridLi, CK=8048590800en_US
dc.identifier.scopusauthoridTsing, NK=6602663351en_US

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