Linear Maps Transforming the Unitary Group

Abstract

AbstractLet U(n) be the group of n × n unitary matrices. We show that if ϕ is a linear transformation sending U(n) into U(m), then m is a multiple of n, and ϕ has the formfor some V, W ∈ 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.