Spectral Theorem

Let  be a Hilbert space,  the set of bounded linear operators from  to itself,  an operator on , and  the operator spectrum of . Then if  and  is normal, there exists a unique resolution of the identity  on the Borel subsets of  which satisfies

Furthermore, every projection  commutes with every  that commutes with .

REFERENCES:

Rudin, W. Theorem 12.23 in Functional Analysis, 2nd ed. New York: McGraw-Hill, 1991.