Past exam of the analysis course of the University of Oxford / 2011 / 6 / c / Solution

If the real part vanishes identically, the norm-difference identity becomes equality, soThus the same substitution as above gives $\Vert Sz\Vert =\Vert z\Vert$ for every $z$. To prove surjectivity as well, observe that both $T$ and $-T$ satisfy the nonnegative-real-part hypothesis. The preceding part makes both $I+T$ and $I-T$ invertible. Their product $S=(I-T)(I+T{)}^{-1}$ is therefore invertible and onto. It is a surjective norm-preserving linear operator, so the equivalences in (a) prove