Let $V$ be a complex inner product space and $T$ be a linear map on $V$ such that $(Tx,Ty)=0$ if $(x,y)=0$. Show that $T=kU$, where $U$ is a unitary map on $V$.
Unfortunately I don`t even know how to approach this problem. I thought about the following:
- show directly that $T^*T=k^2I$
- by the polar decomposition $T=UN$, where $N$ is some non-negative matrix, so one may try to show that $N=I$ in that case
- show that $T$ is an injective (one-to-one) isometry, then it`s unitary
What should I do in this problem? Any hints or solutions are welcomed.