Birkhoff–James Orthogonality: Characterizations, Preservers, and Orthogonality Graphs

Ljiljana Arambašić, Alexander Guterman, Bojan Kuzma, Svetlana Zhilina

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

1 Scopus citations


We present Birkhoff–James orthogonality from historical perspectives to the current development. We compare it with some other orthogonalities, present its properties and its applications, and review the characterizations of Birkhoff–James orthogonality in classical Banach spaces like (ℋ), C-algebras, Hilbert C-modules, or the space of rectangular matrices normed with Schatten norms. We also present the results on characterizations of preservers of Birkhoff–James orthogonality and, by devising a directed graph of the relation, show that in smooth spaces it can completely determine the norm up to (conjugate) linear isometry. Most, though not all, of the results that we state are supplied with (sketches of) the proof.

Original languageEnglish
Title of host publicationTrends in Mathematics
PublisherSpringer Science and Business Media Deutschland GmbH
Number of pages48
StatePublished - 2022
Externally publishedYes

Publication series

NameTrends in Mathematics
ISSN (Print)2297-0215
ISSN (Electronic)2297-024X

Bibliographical note

Publisher Copyright:
© 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.


  • Birkhoff–James orthogonality
  • C-algebra
  • Clique
  • Graph
  • Hilbert C-module
  • Normed vector space
  • Preservers


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