2.6.7 Solution herm-g
Show that if is a Hermitian operator, then so is . As a result, under the conditions of the previous question, is a Hermitian operator too. (And so is just , of course, but is the one with the positive eigenvalues, the squares of the eigenvalues of .)
To show that is Hermitian, just move the two operators to the other side of the inner product one by one. As far as the eigenvalues are concerned, each application of to one of its eigenfunctions multiplies by the eigenvalue, so two applications of multiplies by the square eigenvalue.