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Maps on the sphere of the algebras of matrices. (English) Zbl 1361.15029

This short paper is devoted to unitarily invariant matrix norms, in particular to Schatten \(p\)-norms. The author proves that for a surjective isometry \(T\) on the unit sphere of matrices, with respect to the given Schatten norm, there always exists unitary matrices \(U\), \(V\) so that \(U T(A) V\) is either \(A\) itself, or the transposition, or complex conjugate, or the Hermitian transpose of \(A\).

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A86 Linear preserver problems
46B04 Isometric theory of Banach spaces
15B10 Orthogonal matrices
15A30 Algebraic systems of matrices
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Full Text: Euclid

References:

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[4] D. Tingley, Isometries of the unit sphere , Geom. Dedicata 22 (1987), 371-378. · Zbl 0615.51005
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