I am studying for a test by working out some proofs, and here I have one that I can't seem to solve:
Let $T$ be a self-adjoint linear operator on an inner product space $V$. Prove that this transformation is unitary:
$$ (T + iI)(T - iI)^{-1} $$
In case there is confusion, $(T - iI)$ is automatically invertible since $T$ being self-adjoint implies that $T$ cannot have complex eigenvalues so $(T - iI)$ is invertible. The textbook states that we are to use a previous problem for this proof stating that
$$ || T(x) \pm ix ||^{2} = || T(x) ||^{2} + || x ||^{2} $$
I understand why this statement here is true, but I am not sure how this can be used to prove the statement above. I have spent time on it by trying to prove that $(T + iI)(T - iI)^{-1} [(T + iI)(T - iI)^{-1}]^{*} = I$, but I get no cancellations, so any insight would be greatly appreciated.