Symmetrized Birkhoff–James orthogonality in arbitrary normed spaces

Ljiljana Arambašić; Alexander Guterman; Bojan Kuzma; Rajna Rajić; Svetlana Zhilina

doi:10.1016/j.jmaa.2021.125203

Symmetrized Birkhoff–James orthogonality in arbitrary normed spaces

Ljiljana Arambašić
, Alexander Guterman
, Bojan Kuzma
, Rajna Rajić
, Svetlana Zhilina

University of Zagreb
Lomonosov Moscow State University
Moscow Center for Fundamental and Applied Mathematics
Moscow Institute of Physics and Technology
University of Primorska
Institute of Mathematics, Physics and Mechanics

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

Graph defined by Birkhoff–James orthogonality relation in normed spaces is studied. It is shown that (i) in a normed space of sufficiently large dimension there always exists a nonzero vector which is mutually Birkhoff–James orthogonal to each among a fixed number of given vectors, and (ii) in nonsmooth norms the cardinality of the set of pairwise Birkhoff–James orthogonal vectors may exceed the dimension of the vector space, but this cardinality is always bounded above by a function of the dimension. It is further shown that any given pair of elements in a normed space can be extended to a finite tuple such that each consecutive elements are mutually Birkhoff–James orthogonal; the exact minimal length of the tuple is also determined.

Original language	English
Article number	125203
Journal	Journal of Mathematical Analysis and Applications
Volume	502
Issue number	1
DOIs	https://doi.org/10.1016/j.jmaa.2021.125203
State	Published - 1 Oct 2021
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2021 Elsevier Inc.

Funding

The authors were fully supported by the Croatian Science Foundation under the project IP-2016-06-1046.The work was financially supported by RSF grant 17-11-01124.The author acknowledges the financial support from the Slovenian Research Agency, ARRS (research core funding No. P1-0222 and No. P1-0285).

Funders	Funder number
Javna Agencija za Raziskovalno Dejavnost RS	P1-0285, P1-0222
Hrvatska Zaklada za Znanost	IP-2016-06-1046
Russian Science Foundation	17-11-01124

Keywords

Birkhoff–James orthogonality
Clique number
Graph diameter
Normed vector space

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10.1016/j.jmaa.2021.125203

Cite this

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title = "Symmetrized Birkhoff{\textendash}James orthogonality in arbitrary normed spaces",

abstract = "Graph defined by Birkhoff{\textendash}James orthogonality relation in normed spaces is studied. It is shown that (i) in a normed space of sufficiently large dimension there always exists a nonzero vector which is mutually Birkhoff{\textendash}James orthogonal to each among a fixed number of given vectors, and (ii) in nonsmooth norms the cardinality of the set of pairwise Birkhoff{\textendash}James orthogonal vectors may exceed the dimension of the vector space, but this cardinality is always bounded above by a function of the dimension. It is further shown that any given pair of elements in a normed space can be extended to a finite tuple such that each consecutive elements are mutually Birkhoff{\textendash}James orthogonal; the exact minimal length of the tuple is also determined.",

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AU - Kuzma, Bojan

AU - Rajić, Rajna

AU - Zhilina, Svetlana

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AB - Graph defined by Birkhoff–James orthogonality relation in normed spaces is studied. It is shown that (i) in a normed space of sufficiently large dimension there always exists a nonzero vector which is mutually Birkhoff–James orthogonal to each among a fixed number of given vectors, and (ii) in nonsmooth norms the cardinality of the set of pairwise Birkhoff–James orthogonal vectors may exceed the dimension of the vector space, but this cardinality is always bounded above by a function of the dimension. It is further shown that any given pair of elements in a normed space can be extended to a finite tuple such that each consecutive elements are mutually Birkhoff–James orthogonal; the exact minimal length of the tuple is also determined.

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Symmetrized Birkhoff–James orthogonality in arbitrary normed spaces

Abstract

Bibliographical note

Funding

Keywords

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