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.



Unitary Operator as a complex valued function
Sounds more like a homework/wikipedia problem and not suitable for here but anyways:


First one should maybe mention 
[Stones Theorem](http://en.wikipedia.org/wiki/Stone%27s_theorem_on_one-parameter_unitary_groups) which says there is a one-to-one correpsondence between strongly continuous unitary one-parameter group $\lbrace U(t)\rbrace\_{t \in \mathbb R}$ and self-adjoint operators $A$ given by $U(t)=\exp(\mathrm i tA)$.


This follows from the more general [Borel function calculus](http://en.wikipedia.org/wiki/Borel_functional_calculus) from which also follows that for $A$ self-adjoint 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 self-adjoint operators and the technichal problems comming with these, I refer to the books of Reed and Simon.


