Linear Maps Transforming the Unitary Group

http://dx.doi.org/10.4153/CMB-2003-005-8
Canad. Math. Bull. 46(2003), 54-58

  Published:2003-03-01
 Printed: Mar 2003

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.