In his theoretical physics FAQ, and in specific, http://www.mat.univie.ac.at/~neum/physfaq/topics/mResults" [Broken] page, Neumaier (N henceforth. The name is too big!) says that How is this valid? How can we throw away information and still be A-OK? Ballentine says something similar in Chapter 2 of his book. On pages 49-50, he argues that the essential properties of operators is that they have a spectral representation. He says that people often (wrongly) argue that operators are hermitian because they measure real values. He then goes on to talk about the arbitrariness of convention and states something similar to what is said above. Although I agree with Ballentine that using real numbers to motivate operators is wrong, I still grapple with this hermitian operator business. Can any operator with a spectral representation, not necessarily hermitian, then be chosen to be an operator? I shouldn't think so, but I argue that that would also require the use of convention to restrict the eigenvalues to be real. I would also like to discuss the following: Is this always true? Can you expand a little on this?