Definition 43.8.5 (Cayley Transform).label Let $H$ be a complex Hilbert space, $T: D(T) \to H$ be a densely defined, self-adjoint operator, then
\[U(T): H \to H \quad U(T) = (T - i)(T + i)^{-1}\]
is a unitary operator, known as the Cayley transform of $T$.
Proof. By Lemma 43.8.4, $(T+i)^{-1}: H \to D(T)$ and $(T - i): D(T) \to H$ are well-defined linear bijections, so $U(T)$ is a well-defined linear isomorphism. For any $\xi \in D(T)$, (1) of Lemma 43.8.4 implies that $\norm{(T + i)\xi}_{H} = \norm{(T - i)\xi}_{H}$. As $(T+i)^{-1}$ and $(T - i)$ are both bijections, $U(T)$ is an isometry. Therefore $U(T)$ is unitary.$\square$
Post a Comment