The spectral theorem for unbounded normal operators

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self adjoint operator has a unique positive self adjoint square root A1/2; and an arbitrary self adjoint operator has a unique representation = A+ ~ A~ with A+ and A self adjoint and positive and the range of each contained in the null space of the other. Write ( = (T*Γ) 1/2 and, for complex λ and r ^ 0, let E(\, r) be the null space projection of (| T — XI — r/) +. For compact subsets K of the complex plane E(K)= Λ and for any Borel set M, E{M) = V {E(K): K compact and K £ M) . It is shown that E is the unique spectral measure such that T = \xE(dX) .