A formula for a unitary matrix

Question

Proposition: U = (H+iI)(H-iI)^-1 is unitary if H is self-adjoint. I'm having trouble finding a proof. It's straightforward if (H-iI) and (H+iI)^-1 commute. But I don't see it.

Answer 1

Actually while it's straightward to establish $H+iI$ and $H-iI$ commute, it's unnecessary. To show $\|(H+iI)(H-iI)^{-1}x\|=\|x\|$ for all $x$, since $H-iI$ is invertible (as $H$ is self-adjoint, $i$ is not in its spectrum), it suffices to show $$\forall y, \|(H+iI)(H-iI)^{-1}(H-iI)y\|=\|(H-iI)y\|$$ $$\|(H+iI)y\|=\|(H-iI)y\|$$