Solution
= Solution
Let $D=T^*T-TT^*$. This operator is <self-adjoint>, and
$$
\langle Dx,x\rangle=\|Tx\|^2-\|T^*x\|^2.
$$
If condition (iii) holds, this <quadratic form> vanishes. For a real <inner product>, <polarization identity>[polarization] gives
$$
2\langle Dx,y\rangle=\langle D(x+y),x+y\rangle-\langle Dx,x\rangle-\langle Dy,y\rangle=0.
$$
Therefore $D=0$, which is condition (i). This supplies the nontrivial reverse implication needed for the equivalence.