Geodesic
Linear Maps Transforming the Unitary Group
Canadian mathematical bulletin, Tome 46 (2003) no. 1, pp. 54-58

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DOI

Let $U(n)$ be the group of $n\,\times \,n$ unitary matrices. We show that if $\phi $ is a linear transformation sending $U(n)$ into $U(m)$ , then $m$ is a multiple of $n$ , and $\phi $ has the form $$A\,\mapsto \,V[(A\,\otimes \,{{I}_{s}})\,\otimes \,({{A}^{t}}\,\otimes \,{{I}_{r}})]W$$ for some $V,\,W\,\in \,U(m)$ . From this result, one easily deduces the characterization of linear operators that map $U(n)$ into itself obtained by Marcus. Further generalization of the main theorem is also discussed.
DOI : 10.4153/CMB-2003-005-8
Mots-clés : 15A04, linear map, unitary group, general linear group 
Cheung, Wai-Shun; Li, Chi-Kwong. Linear Maps Transforming the Unitary Group. Canadian mathematical bulletin, Tome 46 (2003) no. 1, pp. 54-58. doi: 10.4153/CMB-2003-005-8
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     year = {2003},
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     doi = {10.4153/CMB-2003-005-8},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2003-005-8/}
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