Two projections $P,Q$ are unitarily equivalent if and only if $$dim(randP)=dim(ranQ)$$ $$dim(kerP)=dim(kerQ)$$
How can we show this? One directionn seems easy:
If $P$ and $Q$ are unitarily eqv, then $PU=UQ$ where $U$ is an isomorphism,hence preserves the dimension of the Hilbert space. Therefore, $dimranP=dimranQ$.
How to show the other direction? Please help me with this.