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A new geometric proof of the spectral theorem for unbounded self-adjointoperators A in a Hilbert space H is given based on a splitting of A in positiveand negative parts A+ and A-. For both operators A+ and A- the spectral familycan be defined immediately and then put together to become the spectral familyof A.

A second look at "A Geometric Proof of the Spectral Theorem for Unbounded Self-Adjoint Operators"