Ok, it's well known, but then I took a review of the notes to my course of quantum information and i found this not proven proposition:

Well, I'm not able to arrange a sounded proof of this simple proposition: i guess i have to choose a base and decompose spectrally the operator but i'm not able to go on. could someone be so kind to show me how to do or at least to suggest me a book or a paper where i could find a proof?