A book on Quantum Mechanics states, "A unitary operator can be considered to be a complex valued function of a Hermitian operator."
Please give a hint on how to prove this assertion.
A book on Quantum Mechanics states, "A unitary operator can be considered to be a complex valued function of a Hermitian operator." Please give a hint on how to prove this assertion. 


Sounds more like a homework/wikipedia problem and not suitable for here but anyways: First one should maybe mention Stones Theorem which says there is a onetoone correpsondence between strongly continuous unitary oneparameter group $\lbrace U(t)\rbrace_{t \in \mathbb R}$ and selfadjoint operators $A$ given by $U(t)=\exp(\mathrm i tA)$. This follows from the more general Borel function calculus from which also follows that for $A$ selfadjoint and $f$ a complex Borel functions with $f=1$ follows that $f(A)$ is a unitary. For a hermitian operator this statement is wrong, but in physics literature there is often not made a difference between hermitian and selfadjoint operators and the technichal problems comming with these, I refer to the books of Reed and Simon. 


Consider $a=i\cdot\log(u)$ by Borel functional calculus. $a$ is selfadjoint since $a^{*}=i\log(u^{*})=i\log(u^{1})=i\log(u)$ and $\exp(ia)=u$. 

