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It is literally (mathematically) false. E.g., the observable defined by the operator $|X|$ cannot be written that way. This observable has also a direct operational definition: I measure the position of the particle on the real line and next I take the absolute value if the outcome. However, it is true from the point of view of Von Neumann algebra theory. ...


2

Your statement is basically true for the conventional Hilbert space without spin or other bells and whistles. I don't wish to get into proofs, as flakey proofs are an excuse to miss the point in such contexts, more often than not, and reduce the issue to their assumptions, better sheltered from your inquisitiveness. You may appreciate this is true and ...


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