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# Linear Maps Transforming the Unitary Group

Published:2003-03-01
Printed: Mar 2003
• Wai-Shun Cheung
• Chi-Kwong Li
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## Abstract

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 $$A \mapsto V[(A\otimes I_s)\oplus (A^t \otimes I_{r})]W$$ 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.
 Keywords: linear map, unitary group, general linear group
 MSC Classifications: 15A04 - Linear transformations, semilinear transformations