= Cayley transform of a bounded accretive operator
{c}
{title2=$S=(I-T)(I+T)^{-1}$}
For a bounded <accretive operator>, $I+T$ is invertible and the transform satisfies $\|S\|\leq1$. The identity $\|(I+T)x\|^2-\|(I-T)x\|^2=4\operatorname{Re}\langle Tx,x\rangle$ 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>.
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