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Linear Maps Transforming the Unitary Group
Published online by Cambridge University Press: 20 November 2018
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
$$A\,\mapsto \,V[(A\,\otimes \,{{I}_{s}})\,\otimes \,({{A}^{t}}\,\otimes \,{{I}_{r}})]W$$
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.
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- Copyright © Canadian Mathematical Society 2003
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