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Adjacency Preserving Maps on Hermitian Matrices

Published online by Cambridge University Press:  20 November 2018

Wen-ling Huang  and
Peter Šemrl
Wen-ling Huang
Affiliation:
Fachbereich Mathematik, Schwerpunkt GD, Universität Hamburg, D-20146 Hamburg, Germany e-mail:huang@math.uni-hamburg.de
Peter Šemrl
Affiliation:
Department of Mathematics, University of Ljubljana, SI-1000 Ljubljana, Slovenia e-mail:peter.semrl@fmf.uni-lj.si
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Abstract

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Hua’s fundamental theorem of the geometry of hermitian matrices characterizes bijective maps on the space of all $n\times n$ hermitianmatrices preserving adjacency in both directions. The problem of possible improvements has been open for a while. There are three natural problems here. Do we need the bijectivity assumption? Can we replace the assumption of preserving adjacency in both directions by the weaker assumption of preserving adjacency in one direction only? Can we obtain such a characterization formaps acting between the spaces of hermitian matrices of different sizes? We answer all three questions for the complex hermitian matrices, thus obtaining the optimal structural result for adjacency preserving maps on hermitian matrices over the complex field.

Keywords

15A0315A0415A5715A99rankadjacency preserving maphermitian matrixgeometry of matrices
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 5 , October 2008 , pp. 1050 - 1066
Copyright
Copyright © Canadian Mathematical Society 2008

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