Cayley transform of a bounded accretive operator ($S=(I-T)(I+T{)}^{-1}$)

For a bounded accretive operator, $I+T$ is invertible and the transform satisfies $\Vert S\Vert \le 1$. The identity $\Vert (I+T)x{\Vert }^2-\Vert (I-T)x{\Vert }^2=4\mathrm{Re}⟨Tx,x⟩$ proves contraction after the substitution $z=(I+T)x$. If the real part vanishes for every $x$, this identity proves norm preservation. Both $T$ and $-T$ are then accretive, making both $I+T$ and $I-T$ invertible; hence $S$ is onto and is a unitary operator.