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The Spectral Theorem of Unbounded Operators

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Proposition

Let be a measure space with a finite measure. Suppose that is a measurable, real-valued function on which is finite a.e. Then the operator on with domain is self-adjoint and is the essential range of .

Spectral Theorem - Multiplication Operator Form

Let be a self-adjoint operator on a separable Hilbert space . Then there exists a measure space with finite, a unitary operator , and a real-valued function on which is finite a.e. such that

  • if and only if ;
  • If , then .

The Functional Calculus Form

Spectral Theorem - Functional Calculus Form

Let be a self-adjoint operator on a separable Hilbert space . Then there exists a unique map such that

  • is a -homomorphism;
  • is norm continuous. i.e., ;
  • If is a sequence of bounded Borel functions with for each and for all and . Then, for any , .
  • If pointwise and if the sequence is bounded, then strongly.

The Projection Valued Measure Form

Projection Valued Measure

Spectral Theorem - Projection Valued Measure Form

There exists a one-to-one correspondence between self-adjoint operators on a separable Hilbert space and projection valued measures on :

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