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Linear Maps Transforming the Unitary Group

Published online by Cambridge University Press:  20 November 2018

Wai-Shun Cheung  and
Chi-Kwong Li
Wai-Shun Cheung
Affiliation:
Centro de Estruturas Lineares e Combinatórias, Universidade de Lisboa, av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal
Chi-Kwong Li
Affiliation:
Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187, U.S.A., email: ckli@math.wm.edu
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Abstract

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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.

Keywords

15A04linear mapunitary groupgeneral linear group

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 46 , Issue 1 , 01 March 2003 , pp. 54 - 58
Copyright
Copyright © Canadian Mathematical Society 2003

References

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