If A is a Hermitian operator, and [A,B]=0, must B necessarily be Hermitian as well?
if Y is an eigenstate of both A and B with respective eigenvalues a and b and respective adjoints (A+) and (B+),
<Y|AB|Y> = <Y|BA|Y>
= <Y|Ab|Y> = <(B+)Y|A|Y>
= b<Y|A|Y> = (b*)<Y|A|Y>
Therefore, b=(b*), and so it follows that B=(B+), or B is Hermitian.