I know that any unitary operator U can be realized in terms of some Hermitian operator K (see equation in #2), and it seems to me that it should also be true that, starting from any Hermitian operator K, the operator defined from that equation exists and is unitary.
Letting M# be the adjoint of M (I can't find a dagger), and * be multiplication,
unitary: M#*M = M*M# = I (Identity operator)
The Attempt at a Solution
Given a Hermitian operator K, define U as above, and then one must prove that exp(iK)*(exp(iK))# = I. That is, that (exp(iK))# =exp(-iK)
Obviously I must use the fact that K=K# to do so, but this only makes the problem to show that exp(-iK#) = (exp(iK))# , which doesn't seem to get me very far. The fact that (M#)-1=(M-1)# also seems tempting to apply somewhere, but I do not see where. Oh, of course there is also the possibility that my conjecture is wrong.
Side note: I presume this should go in some Homework rubrik, because although this is not a Homework problem, it seems like something that would be given for homework in an algebra course. Therefore this should probably go under a rubrik for Algebra, but there was no Algebra listing in the Homework menu.