Abstract
AbstractWe 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 \(\mathbb B(\mathcal {H})\), 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.KeywordsNormed vector spaceBirkhoff–James orthogonality C ∗-algebraHilbert C ∗-modulePreserversGraphClique
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