Unitary operator times projection operator

Question

In this paper, the authors claim that for $C$ a unitary operator and $P$ a projection operator, if $CP \propto P$, then the constant of proportionality must be one. I don't see why this must be the case, can't $C$ be $\theta \cdot I$ for some $\theta$, so that the action isn't completely trivial?

Answer 1

The conclusion is not true, as for $C=iI$ we get $CP=iP.$ However it can be shown that the proportionality constant has absolute value equal $1.$ Indeed, assume $CP= aP.$ Let $Pv=v,$ i.e.$v\in {\rm Im}\, P.$ Hence $Cv=av.$ Since $C$ is a unitary operator then $|a|=1.$