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1. CMB 2003 (vol 46 pp. 54)

Cheung, Wai-Shun; Li, Chi-Kwong
Linear Maps Transforming the Unitary Group
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
Category:15A04

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