Linear Maps Transforming the Unitary Group

[摘要] Let $U(n)$ be the group of $nimes n$ unitary matrices. We show that if$phi$ is a linear transformation sending $U(n)$ into $U(m)$, then $m$ isa multiple of $n$, and $phi$ has the form$$A mapsto V[(Aotimes I_s)oplus (A^t otimes I_{r})]W$$for some $V, W in U(m)$. From this result, one easily deduces thecharacterization of linear operators that map $U(n)$ into itself obtainedby Marcus. Further generalization of the main theorem is also discussed.

[学科分类] 数学（综合）

[关键词] linear map;unitary group;general linear group