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Spectral Theorem


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

 T=int_(sigma(T))lambdadE(lambda).

Furthermore, every projection E(omega) commutes with every S in B(H) that commutes with T.


See also

Operator Spectrum

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References

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

Referenced on Wolfram|Alpha

Spectral Theorem

Cite this as:

Weisstein, Eric W. "Spectral Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SpectralTheorem.html

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