The Spectral Theorem of Unbounded Operators

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.
  

e.g. If is an unbounded positive self-adjoint operator on . Then

The Projection Valued Measure Form

Projection Valued Measure

Let be the Borel measurable space. A projection-valued measure on a separable Hilbert space is a map such that is an orthogonal projection, , , and for any countable collection of disjoint Borel sets, we have where the sum converges in the strong operator topology.

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 :